ASPBE Lean Blueprint

10.2. QuantumBlockEncoding/GHL2025.lean🔗

403 explicit public declarations, in source order.

Definition10.2.1
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “one term robin parameters”. A proposition-valued field is a requirement until a constructor supplies it.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:18. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.1●1 definition
  • structure(4 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.OneTermRobinParameters : Type
    structure QuantumBlockEncoding.GHL2025.OneTermRobinParameters :
  Type

    Fields

    n : ℕ
    kappa : ℕ
    functionPieces : ℕ
    polynomialDegreeCost : ℕ
Definition10.2.2
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “is bulk row”. Classical specification of the indicator oracle U_indic(K1,K2).

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Classical specification of the indicator oracle U_indic(K1,K2). Returns 'true' when row index 'i' is in the bulk region [K1, K2], meaning U_indic maps |i⟩|0⟩ → |i⟩|1⟩. Returns 'false' for boundary rows (0 ≤ i < K1 or K2 < i), meaning U_indic maps |i⟩|0⟩ → |i⟩|0⟩. main.tex:1060-1065 -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:32. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.2●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.isBulkRow (K1 K2 i : ℕ) : Bool
    def QuantumBlockEncoding.GHL2025.isBulkRow
  (K1 K2 i : ℕ) : Bool
    Classical specification of the indicator oracle U_indic(K1,K2).
Returns `true` when row index `i` is in the bulk region [K1, K2],
meaning U_indic maps |i⟩|0⟩ → |i⟩|1⟩.
Returns `false` for boundary rows (0 ≤ i < K1 or K2 < i),
meaning U_indic maps |i⟩|0⟩ → |i⟩|0⟩.
main.tex:1060-1065 -
Definition10.2.3
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “is boundary row”. Complement of isBulkRow: returns true for boundary rows (j < K1 or K2 < j).

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Complement of isBulkRow: returns true for boundary rows (j < K1 or K2 < j). The paper's boundary set is {0,...,K1-1} union {K2+1,...,2^n-1}. main.tex:1113, 1035-1038 -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:39. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.3●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.isBoundaryRow (K1 K2 _gridSize j : ℕ) :
  Bool
    def QuantumBlockEncoding.GHL2025.isBoundaryRow
  (K1 K2 _gridSize j : ℕ) : Bool
    Complement of isBulkRow: returns true for boundary rows (j < K1 or K2 < j).
The paper's boundary set is {0,...,K1-1} union {K2+1,...,2^n-1}.
main.tex:1113, 1035-1038 -
Definition10.2.4
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “robin register partition”. A proposition-valued field is a requirement until a constructor supplies it. Detailed register partition matching the wavefunction ket labels in Eq.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Detailed register partition matching the wavefunction ket labels in Eq. ROBIN clarified (main.tex:1113-1117). Each field is the qubit count for one register in the circuit. Total signal qubits = m_f + 1 + ceil(log2 kappa) + 4 (indicator + ancilla + 1), plus n system qubits. Pure ancillas appear in two groups totaling 2n. figure:1_term_ROBIN caption -

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:49. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.4●1 definition
  • structure(6 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.RobinRegisterPartition : Type
    structure QuantumBlockEncoding.GHL2025.RobinRegisterPartition :
  Type
    Detailed register partition matching the wavefunction ket labels
in Eq. ROBIN clarified (main.tex:1113-1117).
Each field is the qubit count for one register in the circuit.
Total signal qubits = m_f + 1 + ceil(log2 kappa) + 4 (indicator + ancilla + 1),
plus n system qubits. Pure ancillas appear in two groups totaling 2n.
figure:1_term_ROBIN caption -

    Fields

    mfQubits : ℕ
    m_f = ceil(log2 n) + ceil(log2 G_f) + 3 qubits for function oracle O_f.
main.tex:1141 -
    indicatorQubit : ℕ
    1 indicator ancilla qubit set by U_indic. main.tex:1060-1065, 1113 -
    sparseIndexQubits : ℕ
    ceil(log2 kappa) qubits for sparse index s. main.tex:1113 -
    odPureAncillaQubits : ℕ
    n - ceil(log2 kappa) qubits used as pure ancillas for O_D^BS register.
main.tex:1113, 1149 -
    systemQubits : ℕ
    n system qubits for row index j. main.tex:1113 -
    ancillaQubit : ℕ
    1 ancilla qubit (also pure ancilla after cleanup). main.tex:1113, 1159 -
Definition10.2.5
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “total qubits”. Total qubits used by the register partition (all registers summed).

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Total qubits used by the register partition (all registers summed).

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:69. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.5●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.RobinRegisterPartition.totalQubits
  (rp : QuantumBlockEncoding.GHL2025.RobinRegisterPartition) : ℕ
    def QuantumBlockEncoding.GHL2025.RobinRegisterPartition.totalQubits
  (rp :
    QuantumBlockEncoding.GHL2025.RobinRegisterPartition) :
  ℕ
    Total qubits used by the register partition (all registers summed).
Definition10.2.6
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “default robin register partition”. Default register partition from concrete parameters.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Default register partition from concrete parameters. figure:1_term_ROBIN -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:76. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.6●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.defaultRobinRegisterPartition
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GHL2025.RobinRegisterPartition
    def QuantumBlockEncoding.GHL2025.defaultRobinRegisterPartition
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GHL2025.RobinRegisterPartition
    Default register partition from concrete parameters.
figure:1_term_ROBIN -
Definition10.2.7
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “total pure ancillas”. Pure ancilla qubits visible in the Eq.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Pure ancilla qubits visible in the Eq. ROBIN register partition: '(n - ceil(log2 kappa)) + 1' from the O_D^BS register plus the trailing ancilla. This is intentionally narrower than the theorem's full '2n' pure-ancilla budget. The theorem-level count in 'oneTermRobinLayout' and 'oneTermRobinResourceExpr' also includes internal workspace required by the banded sparse-access and oracle subcircuits. Keep this distinction explicit to avoid treating the ket-level register partition as the full resource proof. figure:1_term_ROBIN caption, main.tex:1149, main.tex:1131-1136

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:96. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.7●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.RobinRegisterPartition.totalPureAncillas
  (rp : QuantumBlockEncoding.GHL2025.RobinRegisterPartition) : ℕ
    def QuantumBlockEncoding.GHL2025.RobinRegisterPartition.totalPureAncillas
  (rp :
    QuantumBlockEncoding.GHL2025.RobinRegisterPartition) :
  ℕ
    Pure ancilla qubits visible in the Eq. ROBIN register partition:
`(n - ceil(log2 kappa)) + 1` from the O_D^BS register plus the trailing
ancilla.

This is intentionally narrower than the theorem's full `2n` pure-ancilla
budget.  The theorem-level count in `oneTermRobinLayout` and
`oneTermRobinResourceExpr` also includes internal workspace required by the
banded sparse-access and oracle subcircuits.  Keep this distinction explicit to
avoid treating the ket-level register partition as the full resource proof.
figure:1_term_ROBIN caption, main.tex:1149, main.tex:1131-1136
Definition10.2.8
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “one term robin resource expr”. Theorem 1-term Robin resource shape: 'O(sum_g Q_g n log n + kappa n)' gates and '2n' pure ancillas.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Theorem 1-term Robin resource shape: 'O(sum_g Q_g n log n + kappa n)' gates and '2n' pure ancillas.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:103. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.8●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.oneTermRobinResourceExpr :
  QuantumBlockEncoding.AsymptoticResource
    def QuantumBlockEncoding.GHL2025.oneTermRobinResourceExpr :
  QuantumBlockEncoding.AsymptoticResource
    Theorem 1-term Robin resource shape:
`O(sum_g Q_g n log n + kappa n)` gates and `2n` pure ancillas.
Definition10.2.9
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “deviating indices”. Number of deviating (boundary) indices: K1 + 2^n - K2.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Number of deviating (boundary) indices: K1 + 2^n - K2. The paper notes this is O(1) as it depends on the finite-difference accuracy order. main.tex:1092-1095 -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:112. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.9●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.deviatingIndices (K1 K2 gridSize : ℕ) : ℕ
    def QuantumBlockEncoding.GHL2025.deviatingIndices
  (K1 K2 gridSize : ℕ) : ℕ
    Number of deviating (boundary) indices: K1 + 2^n - K2.
The paper notes this is O(1) as it depends on the finite-difference accuracy order.
main.tex:1092-1095 -
Definition10.2.10
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “one term robin precise resource expr”. Precise gate cost formula from the text (main.tex:1088-1089), before absorbing the O(1) boundary deviation count into the Theorem's simplified formula.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Precise gate cost formula from the text (main.tex:1088-1089), before absorbing the O(1) boundary deviation count into the Theorem's simplified formula. 'O(sum_g Q_g n log n + kappa * (K1 + 2^n - K2) * n)' gates. The term 'K1 + 2^n - K2' is the number of deviating rows, which is O(1). main.tex:1088-1089 -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:121. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.10●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.oneTermRobinPreciseResourceExpr :
  QuantumBlockEncoding.AsymptoticResource
    def QuantumBlockEncoding.GHL2025.oneTermRobinPreciseResourceExpr :
  QuantumBlockEncoding.AsymptoticResource
    Precise gate cost formula from the text (main.tex:1088-1089), before absorbing the
O(1) boundary deviation count into the Theorem's simplified formula.
`O(sum_g Q_g n log n + kappa * (K1 + 2^n - K2) * n)` gates.
The term `K1 + 2^n - K2` is the number of deviating rows, which is O(1).
main.tex:1088-1089 -
Theorem10.2.11
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “deviating indices example”; the hypotheses and conclusion in the code panel fix its exact scope. deviatingIndices computes K1 + gridSize - K2, the number of boundary rows.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. deviatingIndices computes K1 + gridSize - K2, the number of boundary rows. For the fourth-order stencil with K1=2, K2=gridSize(n)-3, this gives 2+3=5. main.tex:1092-1095 -

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:130. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.11●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.deviatingIndices_example :
  QuantumBlockEncoding.GHL2025.deviatingIndices 2
      (QuantumBlockEncoding.gridSize 3 - 3)
      (QuantumBlockEncoding.gridSize 3) =
    5
    theorem QuantumBlockEncoding.GHL2025.deviatingIndices_example :
  QuantumBlockEncoding.GHL2025.deviatingIndices
      2
      (QuantumBlockEncoding.gridSize 3 -
        3)
      (QuantumBlockEncoding.gridSize 3) =
    5
    deviatingIndices computes K1 + gridSize - K2, the number of boundary rows.
For the fourth-order stencil with K1=2, K2=gridSize(n)-3, this gives 2+3=5.
main.tex:1092-1095 -
Definition10.2.12
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “one term robin resource”. Numeric resource useful for concrete search runs with fixed parameters.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Numeric resource useful for concrete search runs with fixed parameters.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:134. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.12●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.oneTermRobinResource
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.Resource
    def QuantumBlockEncoding.GHL2025.oneTermRobinResource
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.Resource
    Numeric resource useful for concrete search runs with fixed parameters.
Theorem10.2.13
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin pure ancilla”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:140. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.13●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobin_pureAncilla
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  (QuantumBlockEncoding.GHL2025.oneTermRobinResource p).pureAncilla =
    2 * p.n
    theorem QuantumBlockEncoding.GHL2025.oneTermRobin_pureAncilla
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  (QuantumBlockEncoding.GHL2025.oneTermRobinResource
        p).pureAncilla =
    2 * p.n
Definition10.2.14
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “one term robin layout”. Register layout for the one-term Robin block encoding.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Register layout for the one-term Robin block encoding. Signal qubits = ⌈log₂ n⌉ + ⌈log₂ G_f⌉ + ⌈log₂ κ⌉ + 4 match the paper's Theorem (main.tex:1098-1109). System qubits address 'n' grid points; pure ancillas are workspace.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:149. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.14●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.oneTermRobinLayout
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.RegisterLayout
    def QuantumBlockEncoding.GHL2025.oneTermRobinLayout
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.RegisterLayout
    Register layout for the one-term Robin block encoding.
Signal qubits = ⌈log₂ n⌉ + ⌈log₂ G_f⌉ + ⌈log₂ κ⌉ + 4
match the paper's Theorem (main.tex:1098-1109).
System qubits address `n` grid points; pure ancillas are workspace.
Definition10.2.15
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “one term robin circuit”. Placeholder circuit for the one-term Robin block encoding.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Placeholder circuit for the one-term Robin block encoding. Gate order matches Fig. 1_term_ROBIN (main.tex:1125-1163): 1. U_indic sets bulk/boundary indicator ancilla. 2. O_DT^S encodes D^T amplitudes (bulk) via sparse-amplitude oracle. 3. Ry_boundary applies controlled rotations for boundary entries. 4. O_D^BS is the banded-sparse-access oracle for D. 5. O_f encodes f(x_j) via amplitude oracle. 6. SWAP between two n-qubit registers. 7. (O_D^BS)^† uncomputes the sparse-access register. Oracle names match 'defaultRobinCircuitSkeleton' field values. figure:1_term_ROBIN -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:166. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.15●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.oneTermRobinCircuit :
  QuantumBlockEncoding.Circuit
    def QuantumBlockEncoding.GHL2025.oneTermRobinCircuit :
  QuantumBlockEncoding.Circuit
    Placeholder circuit for the one-term Robin block encoding.
Gate order matches Fig. 1_term_ROBIN (main.tex:1125-1163):
  1. U_indic sets bulk/boundary indicator ancilla.
  2. O_DT^S encodes D^T amplitudes (bulk) via sparse-amplitude oracle.
  3. Ry_boundary applies controlled rotations for boundary entries.
  4. O_D^BS is the banded-sparse-access oracle for D.
  5. O_f encodes f(x_j) via amplitude oracle.
  6. SWAP between two n-qubit registers.
  7. (O_D^BS)^† uncomputes the sparse-access register.
Oracle names match `defaultRobinCircuitSkeleton` field values.
figure:1_term_ROBIN -
Definition10.2.16
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “one term robin theorem facing fig 4 circuit”. Theorem-facing Fig.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Theorem-facing Fig. 1-term Robin transcript. This label list is the source-facing circuit map used by the conversion window. It deliberately differs from the active seven-gate backend list: the backend matrix product still uses 'oneTermRobinCircuit', while this transcript keeps the sparse-register preparation sides, the explicit 'U_indic^dagger' cleanup slot, and the pre-SWAP 'O_DT^BS' label visible for paper audit. figure:1_term_ROBIN, eq:arbitrary sparcity -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:185. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.16●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.oneTermRobinTheoremFacingFig4Circuit :
  QuantumBlockEncoding.Circuit
    def QuantumBlockEncoding.GHL2025.oneTermRobinTheoremFacingFig4Circuit :
  QuantumBlockEncoding.Circuit
    Theorem-facing Fig. 1-term Robin transcript.

This label list is the source-facing circuit map used by the conversion window.
It deliberately differs from the active seven-gate backend list: the backend
matrix product still uses `oneTermRobinCircuit`, while this transcript keeps
the sparse-register preparation sides, the explicit `U_indic^dagger` cleanup
slot, and the pre-SWAP `O_DT^BS` label visible for paper audit.
figure:1_term_ROBIN, eq:arbitrary sparcity -
Theorem10.2.17
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin theorem facing fig 4 circuit gate list”; the hypotheses and conclusion in the code panel fix its exact scope. The theorem-facing transcript exposes the source-correction slots explicitly.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The theorem-facing transcript exposes the source-correction slots explicitly. This theorem is a transcript guard only. It does not replace 'oneTermRobinCircuit', does not change 'oneTermRobinGateMatrixPlaceholders', and does not promote any oracle correctness or unitarity flag.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:205. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.17●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinTheoremFacingFig4Circuit_gateList :
  QuantumBlockEncoding.GHL2025.oneTermRobinTheoremFacingFig4Circuit =
    [QuantumBlockEncoding.Gate.oracleCall "H_W^(kappa)",
      QuantumBlockEncoding.Gate.oracleCall "U_indic",
      QuantumBlockEncoding.Gate.oracleCall "O_DT^S",
      QuantumBlockEncoding.Gate.oracleCall "Ry_boundary",
      QuantumBlockEncoding.Gate.oracleCall "O_DT^BS",
      QuantumBlockEncoding.Gate.oracleCall "U_indic^dagger",
      QuantumBlockEncoding.Gate.oracleCall "O_f",
      QuantumBlockEncoding.Gate.swap 0 0,
      QuantumBlockEncoding.Gate.oracleCall "(O_D^BS)^dagger",
      QuantumBlockEncoding.Gate.oracleCall "(H_W^(kappa))^dagger"]
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinTheoremFacingFig4Circuit_gateList :
  QuantumBlockEncoding.GHL2025.oneTermRobinTheoremFacingFig4Circuit =
    [QuantumBlockEncoding.Gate.oracleCall
        "H_W^(kappa)",
      QuantumBlockEncoding.Gate.oracleCall
        "U_indic",
      QuantumBlockEncoding.Gate.oracleCall
        "O_DT^S",
      QuantumBlockEncoding.Gate.oracleCall
        "Ry_boundary",
      QuantumBlockEncoding.Gate.oracleCall
        "O_DT^BS",
      QuantumBlockEncoding.Gate.oracleCall
        "U_indic^dagger",
      QuantumBlockEncoding.Gate.oracleCall
        "O_f",
      QuantumBlockEncoding.Gate.swap 0 0,
      QuantumBlockEncoding.Gate.oracleCall
        "(O_D^BS)^dagger",
      QuantumBlockEncoding.Gate.oracleCall
        "(H_W^(kappa))^dagger"]
    The theorem-facing transcript exposes the source-correction slots explicitly.

This theorem is a transcript guard only.  It does not replace
`oneTermRobinCircuit`, does not change `oneTermRobinGateMatrixPlaceholders`, and
does not promote any oracle correctness or unitarity flag.
Theorem10.2.18
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin active backend circuit gate list”; the hypotheses and conclusion in the code panel fix its exact scope. The active backend circuit remains the seven-gate product currently used by the finite matrix semantics.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The active backend circuit remains the seven-gate product currently used by the finite matrix semantics. Theorem-facing proof maps must not call this list the full Fig. 1-term Robin transcript, because it omits both 'H_W^(kappa)' sides and the explicit 'U_indic^dagger' source slot.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:227. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.18●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinActiveBackendCircuit_gateList :
  QuantumBlockEncoding.GHL2025.oneTermRobinCircuit =
    [QuantumBlockEncoding.Gate.oracleCall "U_indic",
      QuantumBlockEncoding.Gate.oracleCall "O_DT^S",
      QuantumBlockEncoding.Gate.oracleCall "Ry_boundary",
      QuantumBlockEncoding.Gate.oracleCall "O_D^BS",
      QuantumBlockEncoding.Gate.oracleCall "O_f",
      QuantumBlockEncoding.Gate.swap 0 0,
      QuantumBlockEncoding.Gate.oracleCall "(O_D^BS)^†"]
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinActiveBackendCircuit_gateList :
  QuantumBlockEncoding.GHL2025.oneTermRobinCircuit =
    [QuantumBlockEncoding.Gate.oracleCall
        "U_indic",
      QuantumBlockEncoding.Gate.oracleCall
        "O_DT^S",
      QuantumBlockEncoding.Gate.oracleCall
        "Ry_boundary",
      QuantumBlockEncoding.Gate.oracleCall
        "O_D^BS",
      QuantumBlockEncoding.Gate.oracleCall
        "O_f",
      QuantumBlockEncoding.Gate.swap 0 0,
      QuantumBlockEncoding.Gate.oracleCall
        "(O_D^BS)^†"]
    The active backend circuit remains the seven-gate product currently used by
the finite matrix semantics.

Theorem-facing proof maps must not call this list the full Fig. 1-term Robin
transcript, because it omits both `H_W^(kappa)` sides and the explicit
`U_indic^dagger` source slot.
Definition10.2.19
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “one term robin normalizer”. Symbolic normalizer α = N_D · N_f · κ for the one-term Robin construction.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Symbolic normalizer α = N_D · N_f · κ for the one-term Robin construction.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:239. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.19●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.oneTermRobinNormalizer :
  QuantumBlockEncoding.Coeff
    def QuantumBlockEncoding.GHL2025.oneTermRobinNormalizer :
  QuantumBlockEncoding.Coeff
    Symbolic normalizer α = N_D · N_f · κ for the one-term Robin construction.
Definition10.2.20
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “one term robin spec”. Block-encoding spec for the one-term Robin derivative operator.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Block-encoding spec for the one-term Robin derivative operator. Takes the target matrix as a parameter so the spec is reusable across different stencil choices and boundary data without creating import cycles. Normalizer: symbolic 'N_D · N_f · κ'. Error: zero (exact encoding, no approximation yet).

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:249. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.20●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.oneTermRobinSpec
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (mat :
    QuantumBlockEncoding.Matrix (QuantumBlockEncoding.gridSize p.n)
      (QuantumBlockEncoding.gridSize p.n) QuantumBlockEncoding.Coeff) :
  QuantumBlockEncoding.BlockEncodingSpec QuantumBlockEncoding.Coeff
    (QuantumBlockEncoding.gridSize p.n)
    (QuantumBlockEncoding.gridSize p.n)
    def QuantumBlockEncoding.GHL2025.oneTermRobinSpec
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (mat :
    QuantumBlockEncoding.Matrix
      (QuantumBlockEncoding.gridSize p.n)
      (QuantumBlockEncoding.gridSize p.n)
      QuantumBlockEncoding.Coeff) :
  QuantumBlockEncoding.BlockEncodingSpec
    QuantumBlockEncoding.Coeff
    (QuantumBlockEncoding.gridSize p.n)
    (QuantumBlockEncoding.gridSize p.n)
    Block-encoding spec for the one-term Robin derivative operator.
Takes the target matrix as a parameter so the spec is reusable across different
stencil choices and boundary data without creating import cycles.
Normalizer: symbolic `N_D · N_f · κ`.
Error: zero (exact encoding, no approximation yet).
Theorem10.2.21
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin spec ancilla”; the hypotheses and conclusion in the code panel fix its exact scope. The spec's pure ancilla matches the resource formula.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The spec's pure ancilla matches the resource formula.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:260. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.21●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinSpec_ancilla
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (mat :
    QuantumBlockEncoding.Matrix (QuantumBlockEncoding.gridSize p.n)
      (QuantumBlockEncoding.gridSize p.n) QuantumBlockEncoding.Coeff) :
  (QuantumBlockEncoding.GHL2025.oneTermRobinSpec p
          mat).resource.pureAncilla =
    2 * p.n
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinSpec_ancilla
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (mat :
    QuantumBlockEncoding.Matrix
      (QuantumBlockEncoding.gridSize p.n)
      (QuantumBlockEncoding.gridSize p.n)
      QuantumBlockEncoding.Coeff) :
  (QuantumBlockEncoding.GHL2025.oneTermRobinSpec
          p mat).resource.pureAncilla =
    2 * p.n
    The spec's pure ancilla matches the resource formula.
Theorem10.2.22
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin spec circuit cost”; the hypotheses and conclusion in the code panel fix its exact scope. The spec's circuit local cost: the SWAP placeholder costs 3 CNOTs and each unexpanded oracle call is counted as one unresolved call in the candidate score.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The spec's circuit local cost: the SWAP placeholder costs 3 CNOTs and each unexpanded oracle call is counted as one unresolved call in the candidate score. figure:1_term_ROBIN

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:269. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.22●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinSpec_circuitCost :
  QuantumBlockEncoding.GHL2025.oneTermRobinCircuit.resource =
    QuantumBlockEncoding.Resource.ofCountsWithDepth 0 3 6 0 9
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinSpec_circuitCost :
  QuantumBlockEncoding.GHL2025.oneTermRobinCircuit.resource =
    QuantumBlockEncoding.Resource.ofCountsWithDepth
      0 3 6 0 9
    The spec's circuit local cost: the SWAP placeholder costs 3 CNOTs and each
unexpanded oracle call is counted as one unresolved call in the candidate score.
figure:1_term_ROBIN
Theorem10.2.23
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin normalizer eval”; the hypotheses and conclusion in the code panel fix its exact scope. Evaluating the symbolic normalizer 'N_D · N_f · κ' under an environment gives the product of the three symbol values.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Evaluating the symbolic normalizer 'N_D · N_f · κ' under an environment gives the product of the three symbol values.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:274. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.23●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinNormalizer_eval
  (env : String → ℚ) :
  QuantumBlockEncoding.Coeff.evalWith env
      QuantumBlockEncoding.GHL2025.oneTermRobinNormalizer =
    env "N_D" * env "N_f" * env "kappa"
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinNormalizer_eval
  (env : String → ℚ) :
  QuantumBlockEncoding.Coeff.evalWith env
      QuantumBlockEncoding.GHL2025.oneTermRobinNormalizer =
    env "N_D" * env "N_f" * env "kappa"
    Evaluating the symbolic normalizer `N_D · N_f · κ` under an environment gives
the product of the three symbol values.
Definition10.2.24
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “one term robin claim”. The paper's one-term Robin block-encoding construction claim.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The paper's one-term Robin block-encoding construction claim.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:279. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.24●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.oneTermRobinClaim :
  QuantumBlockEncoding.ConstructionClaim
    def QuantumBlockEncoding.GHL2025.oneTermRobinClaim :
  QuantumBlockEncoding.ConstructionClaim
    The paper's one-term Robin block-encoding construction claim.
Definition10.2.25
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “one dim hamiltonian resource expr”. One-dimensional Hamiltonian block-encoding resource shape.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. One-dimensional Hamiltonian block-encoding resource shape.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:288. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.25●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.oneDimHamiltonianResourceExpr :
  QuantumBlockEncoding.AsymptoticResource
    def QuantumBlockEncoding.GHL2025.oneDimHamiltonianResourceExpr :
  QuantumBlockEncoding.AsymptoticResource
    One-dimensional Hamiltonian block-encoding resource shape.
Definition10.2.26
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “one dim hamiltonian claim”. The paper's 1D Hamiltonian block-encoding construction claim.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The paper's 1D Hamiltonian block-encoding construction claim.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:296. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.26●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.oneDimHamiltonianClaim :
  QuantumBlockEncoding.ConstructionClaim
    def QuantumBlockEncoding.GHL2025.oneDimHamiltonianClaim :
  QuantumBlockEncoding.ConstructionClaim
    The paper's 1D Hamiltonian block-encoding construction claim.
Definition10.2.27
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “multi dim hamiltonian resource expr”. Multidimensional Hamiltonian block-encoding resource shape.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Multidimensional Hamiltonian block-encoding resource shape.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:305. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.27●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.multiDimHamiltonianResourceExpr :
  QuantumBlockEncoding.AsymptoticResource
    def QuantumBlockEncoding.GHL2025.multiDimHamiltonianResourceExpr :
  QuantumBlockEncoding.AsymptoticResource
    Multidimensional Hamiltonian block-encoding resource shape.
Definition10.2.28
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “multi dim hamiltonian claim”. The paper's multidimensional Hamiltonian block-encoding construction claim.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The paper's multidimensional Hamiltonian block-encoding construction claim.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:313. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.28●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.multiDimHamiltonianClaim :
  QuantumBlockEncoding.ConstructionClaim
    def QuantumBlockEncoding.GHL2025.multiDimHamiltonianClaim :
  QuantumBlockEncoding.ConstructionClaim
    The paper's multidimensional Hamiltonian block-encoding construction claim.
Definition10.2.29
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “obligation record”. A proposition-valued field is a requirement until a constructor supplies it. A proof obligation tracked by description and paper source anchor.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. A proof obligation tracked by description and paper source anchor. 'proved' is 'Bool' (not 'Prop') so that unproved obligations are honest data, not mathematically false claims. main.tex -

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:324. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.29●1 definition
  • structure(3 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.ObligationRecord : Type
    structure QuantumBlockEncoding.GHL2025.ObligationRecord :
  Type
    A proof obligation tracked by description and paper source anchor.
`proved` is `Bool` (not `Prop`) so that unproved obligations are honest data,
not mathematically false claims. main.tex -

    Fields

    description : String
    source : String
    proved : Bool
Definition10.2.30
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “robin circuit skeleton”. A proposition-valued field is a requirement until a constructor supplies it. Circuit skeleton matching Fig.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Circuit skeleton matching Fig. 1_term_ROBIN (main.tex:1137-1167). Each field corresponds to a labeled box or operation in the figure. All oracles are recorded as symbolic names; their implementation is delegated to separate oracle-contract structures. figure:1_term_ROBIN -

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:334. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.30●1 definition
  • structure(13 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.RobinCircuitSkeleton : Type
    structure QuantumBlockEncoding.GHL2025.RobinCircuitSkeleton :
  Type
    Circuit skeleton matching Fig. 1_term_ROBIN (main.tex:1137-1167).
Each field corresponds to a labeled box or operation in the figure.
All oracles are recorded as symbolic names; their implementation is delegated
to separate oracle-contract structures. figure:1_term_ROBIN -

    Fields

    indicatorOracle : String
    Bulk/boundary indicator unitary U_indic(K1,K2). main.tex:1088-1099 -
    K1 : ℕ
    Bulk window lower bound K1. main.tex:1095 -
    K2 : ℕ
    Bulk window upper bound K2. main.tex:1095 -
    sparseAmplitudeOracleDT : String
    Sparse-amplitude oracle for transposed derivative D^T.
Lemma 3 (main.tex:822-849). -
    bandedSparseAccessOracleD : String
    Banded-sparse-access oracle for D. Lemma 1 (main.tex:784-801). -
    bandedSparseAccessOracleD_dagger : String
    Hermitian conjugate of the banded-sparse-access oracle (O_D^BS)^†.
figure:1_term_ROBIN caption, main.tex:1148 -
    controlledRyBoundary : String
    Boundary-controlled R_y rotations with angles theta_j^s = arccos(D_j^(s)/N_D).
main.tex:1115-1120 -
    functionOracle : String
    Amplitude oracle O_f for piecewise polynomial f(x). Thm 5 (main.tex:870-910). -
    swapOperation : String
    SWAP between two n-qubit registers. figure:1_term_ROBIN, main.tex:1140 -
    mergeFrame : String
    Y-frame register merge (no quantum operation). figure:1_term_ROBIN caption -
    mfSignalQubits : ℕ
    m_f = ceil(log2 n) + ceil(log2 G_f) + 3 qubits reserved for O_f. main.tex:1141 -
    kappaSignalQubits : ℕ
    ceil(log2 kappa) qubits for sparse indexing. figure:1_term_ROBIN caption -
    pureAncillaODRegister : ℕ
    n - ceil(log2 kappa) pure ancilla qubits for O_D^BS register.
figure:1_term_ROBIN caption, main.tex:1149 -
Definition10.2.31
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “robin gamma 1”. A proposition-valued field is a requirement until a constructor supplies it. Eq.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Eq. ROBIN clarified, gamma_1 component (main.tex:1113). State after U_indic sets the indicator ancilla. The boundary and bulk summation terms have different normalizers: boundary: 1/(N_D · sqrt(kappa)), indicator |0> bulk: 1/sqrt(kappa), indicator |1> main.tex:1113 -

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:373. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.31●1 definition
  • structure(7 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.RobinGamma1 : Type
    structure QuantumBlockEncoding.GHL2025.RobinGamma1 :
  Type
    Eq. ROBIN clarified, gamma_1 component (main.tex:1113).
State after U_indic sets the indicator ancilla.
The boundary and bulk summation terms have different normalizers:
  boundary: 1/(N_D · sqrt(kappa)), indicator |0>
  bulk:     1/sqrt(kappa),        indicator |1>
main.tex:1113 -

    Fields

    kappa : ℕ
    Sparse index upper bound: s in {0, ..., kappa-1}. main.tex:1113 -
    K1 : ℕ
    Bulk window lower bound. main.tex:1095, 1113 -
    K2 : ℕ
    Bulk window upper bound. main.tex:1095, 1113 -
    gridSize : ℕ
    Grid size = 2^n. main.tex:1113 -
    boundaryNormalizer : QuantumBlockEncoding.Coeff
    Boundary normalizer N_D · sqrt(kappa) (symbolic).
The boundary summation in gamma_1 is scaled by 1/(N_D · sqrt(kappa)). main.tex:1113 -
    bulkNormalizer : QuantumBlockEncoding.Coeff
    Bulk normalizer sqrt(kappa) (symbolic).
The bulk summation in gamma_1 is scaled by 1/sqrt(kappa) — notably
without the N_D factor, because the sparse-amplitude oracle has not yet
been applied. main.tex:1113 -
    mfQubits : ℕ
    Number of m_f qubits reserved for O_f. main.tex:1141 -
Definition10.2.32
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “robin gamma 2”. A proposition-valued field is a requirement until a constructor supplies it. Eq.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Eq. ROBIN clarified, gamma_2 component (main.tex:1115). State after sparse-amplitude oracle encodes D^T values. main.tex:1115 -

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:396. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.32●1 definition
  • structure(6 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.RobinGamma2 : Type
    structure QuantumBlockEncoding.GHL2025.RobinGamma2 :
  Type
    Eq. ROBIN clarified, gamma_2 component (main.tex:1115).
State after sparse-amplitude oracle encodes D^T values. main.tex:1115 -

    Fields

    kappa : ℕ
    Sparse index upper bound. main.tex:1115 -
    K1 : ℕ
    Bulk window lower bound. main.tex:1115 -
    K2 : ℕ
    Bulk window upper bound. main.tex:1115 -
    gridSize : ℕ
    Grid size. main.tex:1115 -
    normalizer : QuantumBlockEncoding.Coeff
    Normalization factor N_D * sqrt(kappa) (symbolic). main.tex:1115 -
    hasOrthogonalRemainder : Bool
    Whether this includes the orthogonal remainder "+ ...". main.tex:1115 -
Definition10.2.33
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “robin gamma 3”. A proposition-valued field is a requirement until a constructor supplies it. Eq.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Eq. ROBIN clarified, gamma_3 component (main.tex:1117). State after O_f encodes f(x_j) values. main.tex:1117 -

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:413. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.33●1 definition
  • structure(7 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.RobinGamma3 : Type
    structure QuantumBlockEncoding.GHL2025.RobinGamma3 :
  Type
    Eq. ROBIN clarified, gamma_3 component (main.tex:1117).
State after O_f encodes f(x_j) values. main.tex:1117 -

    Fields

    kappa : ℕ
    Sparse index upper bound. main.tex:1117 -
    K1 : ℕ
    Bulk window lower bound. main.tex:1117 -
    K2 : ℕ
    Bulk window upper bound. main.tex:1117 -
    gridSize : ℕ
    Grid size. main.tex:1117 -
    normalizer : QuantumBlockEncoding.Coeff
    Normalization factor N_D * N_f * kappa (symbolic). main.tex:1117 -
    hasOrthogonalRemainder : Bool
    Whether this includes the orthogonal remainder "+ ...". main.tex:1117 -
    pureAncillaQubits : ℕ
    Number of pure ancilla qubits identified: n - ceil(log2 kappa) + 1. main.tex:1117, 1149 -
Definition10.2.34
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “robin wavefunction decomposition”. A proposition-valued field is a requirement until a constructor supplies it. Bundle of the three intermediate wavefunction states from Eq.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Bundle of the three intermediate wavefunction states from Eq. ROBIN clarified. Captures the full circuit state evolution from input through U_indic, O_DT^S, and O_f. main.tex:1113-1117, figure:1_term_ROBIN -

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:433. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.34●1 definition
  • structure(7 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.RobinWavefunctionDecomposition : Type
    structure QuantumBlockEncoding.GHL2025.RobinWavefunctionDecomposition :
  Type
    Bundle of the three intermediate wavefunction states from Eq. ROBIN clarified.
Captures the full circuit state evolution from input through U_indic, O_DT^S, and O_f.
main.tex:1113-1117, figure:1_term_ROBIN -

    Fields

    gamma1 : QuantumBlockEncoding.GHL2025.RobinGamma1
    gamma_1: state after U_indic. main.tex:1113 -
    gamma2 : QuantumBlockEncoding.GHL2025.RobinGamma2
    gamma_2: state after sparse-amplitude oracle. main.tex:1115 -
    gamma3 : QuantumBlockEncoding.GHL2025.RobinGamma3
    gamma_3: state after function oracle O_f. main.tex:1117 -
    kappa : ℕ
    The shared sparse index upper bound. -
    K1 : ℕ
    The shared bulk window lower bound. -
    K2 : ℕ
    The shared bulk window upper bound. -
    gridSize : ℕ
    The shared grid size 2^n. -
Definition10.2.35
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “default robin wavefunction decomposition”. Default wavefunction decomposition from concrete parameters.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Default wavefunction decomposition from concrete parameters. figure:1_term_ROBIN, main.tex:1113-1117 -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:452. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.35●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.defaultRobinWavefunctionDecomposition
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GHL2025.RobinWavefunctionDecomposition
    def QuantumBlockEncoding.GHL2025.defaultRobinWavefunctionDecomposition
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GHL2025.RobinWavefunctionDecomposition
    Default wavefunction decomposition from concrete parameters.
figure:1_term_ROBIN, main.tex:1113-1117 -
Definition10.2.36
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “robin proof obligations”. A proposition-valued field is a requirement until a constructor supplies it. Bundle of proof obligations for the one-term Robin block encoding.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Bundle of proof obligations for the one-term Robin block encoding. Each obligation references a specific claim in the paper and tracks whether it has been formally proved. None are proved in the current version. Guseynov-Huang-Liu 2025, one-term Robin theorem, arXiv:2506.20478. -

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:488. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.36●1 definition
  • structure(10 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.RobinProofObligations : Type
    structure QuantumBlockEncoding.GHL2025.RobinProofObligations :
  Type
    Bundle of proof obligations for the one-term Robin block encoding.
Each obligation references a specific claim in the paper and tracks whether
it has been formally proved. None are proved in the current version.
Guseynov-Huang-Liu 2025, one-term Robin theorem, arXiv:2506.20478. -

    Fields

    indicatorUnitary : QuantumBlockEncoding.GHL2025.ObligationRecord
    U_indic is unitary and implements the bulk-window predicate. -
    sparseAmplitudeOracleCorrect : QuantumBlockEncoding.GHL2025.ObligationRecord
    Sparse-amplitude oracle O_D^S is unitary and encodes D^(s)/N_D. -
    bandedSparseAccessCorrect : QuantumBlockEncoding.GHL2025.ObligationRecord
    Banded-sparse-access oracle O_D^BS is unitary and writes r_si. -
    functionOracleCorrect : QuantumBlockEncoding.GHL2025.ObligationRecord
    Function oracle O_f is unitary and block-encodes f(x_j)/N_f. -
    controlledRyCorrect : QuantumBlockEncoding.GHL2025.ObligationRecord
    Controlled R_y angles encode boundary D_j^(s)/N_D correctly. -
    circuitUnitary : QuantumBlockEncoding.GHL2025.ObligationRecord
    Full circuit U_Ak^(1) is unitary. -
    blockExtraction : QuantumBlockEncoding.GHL2025.ObligationRecord
    Block extraction: (⟨0|⊗I) U (|0⟩⊗I) = A_k / (N_D * N_f * kappa). -
    resourceBound : QuantumBlockEncoding.GHL2025.ObligationRecord
    Resource bound: O(sum Q_g n log n + kappa n) gates, 2n pure ancillas. -
    ancillaCleanup : QuantumBlockEncoding.GHL2025.ObligationRecord
    Pure ancilla cleanup: 2n ancillas returned to |0⟩. figure:1_term_ROBIN caption -
    ghostPointElimination : QuantumBlockEncoding.GHL2025.ObligationRecord
    Ghost-point elimination yields correct Robin boundary row coefficients. -
Definition10.2.37
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “default robin circuit skeleton”. Default circuit skeleton for the one-term Robin construction, with oracle names matching the paper's notation.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Default circuit skeleton for the one-term Robin construction, with oracle names matching the paper's notation. figure:1_term_ROBIN -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:553. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.37●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.defaultRobinCircuitSkeleton
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GHL2025.RobinCircuitSkeleton
    def QuantumBlockEncoding.GHL2025.defaultRobinCircuitSkeleton
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GHL2025.RobinCircuitSkeleton
    Default circuit skeleton for the one-term Robin construction,
with oracle names matching the paper's notation. figure:1_term_ROBIN -
Definition10.2.38
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “banded sparse access paper contract”. A proposition-valued field is a requirement until a constructor supplies it. Paper-level source contract for the banded sparse-access oracle in Lemma 1.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Paper-level source contract for the banded sparse-access oracle in Lemma 1. The input register is the padded sparse-index register '|0>^(n-l)|s>^l' followed by the row register '|i>^n'; the output is '|r_si>^n|i>^n'. This record is intentionally separate from the current 'bandedSparseAccessMatrix' helper, which overwrites the system register with a Robin column map and therefore does not yet implement this paper contract. Guseynov-Huang-Liu 2025, Lemma 1, arXiv:2506.20478.

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:577. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.38●1 definition
  • structure(15 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.BandedSparseAccessPaperContract : Type
    structure QuantumBlockEncoding.GHL2025.BandedSparseAccessPaperContract :
  Type
    Paper-level source contract for the banded sparse-access oracle in Lemma 1.

The input register is the padded sparse-index register
`|0>^(n-l)|s>^l` followed by the row register `|i>^n`; the output is
`|r_si>^n|i>^n`.  This record is intentionally separate from the current
`bandedSparseAccessMatrix` helper, which overwrites the system register with a
Robin column map and therefore does not yet implement this paper contract.
Guseynov-Huang-Liu 2025, Lemma 1, arXiv:2506.20478.

    Fields

    sourceAnchor : String
    rowRegisterQubits : ℕ
    paddedZeroQubits : ℕ
    sparseIndexQubits : ℕ
    outputAddressQubits : ℕ
    inputKet : String
    outputKet : String
    imageFormula : String
    cleanInputDomain : QuantumBlockEncoding.GHL2025.ObligationRecord
    widthCompatible : QuantumBlockEncoding.GHL2025.ObligationRecord
    addressRange : QuantumBlockEncoding.GHL2025.ObligationRecord
    noSpill : QuantumBlockEncoding.GHL2025.ObligationRecord
    forwardCorrect : QuantumBlockEncoding.GHL2025.ObligationRecord
    daggerCleanup : QuantumBlockEncoding.GHL2025.ObligationRecord
    unitaryExtension : QuantumBlockEncoding.GHL2025.ObligationRecord
Definition10.2.39
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “default banded sparse access paper contract”. Default Lemma 1 register contract for the one-term Robin parameters.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Default Lemma 1 register contract for the one-term Robin parameters. The 'widthCompatible' obligation stays explicit because the current parameter type does not enforce 'clog2 kappa <= n'; faithful proofs should discharge that side condition or specialize to a parameter family where it is available.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:602. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.39●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessPaperContract
    def QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessPaperContract
    Default Lemma 1 register contract for the one-term Robin parameters.

The `widthCompatible` obligation stays explicit because the current parameter
type does not enforce `clog2 kappa <= n`; faithful proofs should discharge that
side condition or specialize to a parameter family where it is available.
Definition10.2.40
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “derivative oracle contract”. A proposition-valued field is a requirement until a constructor supplies it. Contract for the derivative oracle O_D: sparse-access oracle for the banded stencil matrix.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Contract for the derivative oracle O_D: sparse-access oracle for the banded stencil matrix. Records stencil metadata, bandwidth, and a correctness obligation. main.tex:784-801 -

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:650. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.40●1 definition
  • structure(5 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.DerivativeOracleContract (n : ℕ) : Type
    structure QuantumBlockEncoding.GHL2025.DerivativeOracleContract
  (n : ℕ) : Type
    Contract for the derivative oracle O_D: sparse-access oracle for the banded
stencil matrix. Records stencil metadata, bandwidth, and a correctness obligation. main.tex:784-801 -

    Fields

    stencil : QuantumBlockEncoding.Stencil
    bandwidth : ℕ
    matrix : QuantumBlockEncoding.Matrix (QuantumBlockEncoding.gridSize n) (QuantumBlockEncoding.gridSize n)
  QuantumBlockEncoding.Coeff
    sparseCorrect : QuantumBlockEncoding.GHL2025.ObligationRecord
    Obligation: O_D^BS correctly maps sparse indices to matrix entries.
main.tex:784-801 -
    bandwidth_eq : self.bandwidth = self.stencil.width
Definition10.2.41
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “function oracle contract”. A proposition-valued field is a requirement until a constructor supplies it. Contract for the function oracle O_f: amplitude oracle encoding f(x) on the grid.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Contract for the function oracle O_f: amplitude oracle encoding f(x) on the grid. Records the piece count, normalization bound, and a correctness obligation. main.tex:870-910

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:661. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.41●1 definition
  • structure(3 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.FunctionOracleContract (n : ℕ) : Type
    structure QuantumBlockEncoding.GHL2025.FunctionOracleContract
  (n : ℕ) : Type
    Contract for the function oracle O_f: amplitude oracle encoding f(x) on the
grid. Records the piece count, normalization bound, and a correctness obligation. main.tex:870-910

    Fields

    functionPieces : ℕ
    normalizerBound : QuantumBlockEncoding.Coeff
    amplitudeCorrect : QuantumBlockEncoding.GHL2025.ObligationRecord
    Obligation: O_f correctly block-encodes f(x_j)/N_f on the grid.
main.tex:870-910 -
Definition10.2.42
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “derivative oracle resource”. Resource for the derivative oracle O_D using the banded sparse-access formula from Lemma 1 of Guseynov-Huang-Liu 2025.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Resource for the derivative oracle O_D using the banded sparse-access formula from Lemma 1 of Guseynov-Huang-Liu 2025. The half-bandwidth parameter is 'stencil.leftRadius' (assumes a symmetric stencil where leftRadius = rightRadius).

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:671. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.42●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.derivativeOracleResource (n : ℕ)
  (s : QuantumBlockEncoding.Stencil) : QuantumBlockEncoding.Resource
    def QuantumBlockEncoding.GHL2025.derivativeOracleResource
  (n : ℕ)
  (s : QuantumBlockEncoding.Stencil) :
  QuantumBlockEncoding.Resource
    Resource for the derivative oracle O_D using the banded sparse-access formula
from Lemma 1 of Guseynov-Huang-Liu 2025. The half-bandwidth parameter is
`stencil.leftRadius` (assumes a symmetric stencil where leftRadius = rightRadius).
Theorem10.2.43
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “derivative oracle resource pure ancilla”; the hypotheses and conclusion in the code panel fix its exact scope. The derivative oracle's pure ancilla count is n - 1 (from Lemma 1).

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The derivative oracle's pure ancilla count is n - 1 (from Lemma 1).

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:675. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.43●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.derivativeOracleResource_pureAncilla
  (n : ℕ) (s : QuantumBlockEncoding.Stencil) :
  (QuantumBlockEncoding.GHL2025.derivativeOracleResource n
        s).pureAncilla =
    n - 1
    theorem QuantumBlockEncoding.GHL2025.derivativeOracleResource_pureAncilla
  (n : ℕ)
  (s : QuantumBlockEncoding.Stencil) :
  (QuantumBlockEncoding.GHL2025.derivativeOracleResource
        n s).pureAncilla =
    n - 1
    The derivative oracle's pure ancilla count is n - 1 (from Lemma 1).
Definition10.2.44
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “one term robin theorem data”. A proposition-valued field is a requirement until a constructor supplies it. Typed theorem data for Theorem one-term block-encoding (main.tex:1098-1109).

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Typed theorem data for Theorem one-term block-encoding (main.tex:1098-1109). Captures the exact block-encoding tuple (α, m, a) from the paper: α = N_D · N_f · κ (normalizer) m = ⌈log₂ n⌉ + ⌈log₂ G_f⌉ + ⌈log₂ κ⌉ + 4 (signal ancilla qubits) a = 0 (zero approximation error) along with the gate-count and pure-ancilla resource claims.

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:684. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.44●1 definition
  • structure(6 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.OneTermRobinTheoremData : Type
    structure QuantumBlockEncoding.GHL2025.OneTermRobinTheoremData :
  Type
    Typed theorem data for Theorem one-term block-encoding (main.tex:1098-1109).
Captures the exact block-encoding tuple (α, m, a) from the paper:
  α = N_D · N_f · κ    (normalizer)
  m = ⌈log₂ n⌉ + ⌈log₂ G_f⌉ + ⌈log₂ κ⌉ + 4   (signal ancilla qubits)
  a = 0                (zero approximation error)
along with the gate-count and pure-ancilla resource claims.

    Fields

    alpha : QuantumBlockEncoding.Coeff
    Block-encoding normalizer α = N_D · N_f · κ. main.tex:1102 -
    signalQubits : ℕ
    Signal ancilla qubits m = ⌈log₂ n⌉ + ⌈log₂ G_f⌉ + ⌈log₂ κ⌉ + 4. main.tex:1102 -
    error : QuantumBlockEncoding.Coeff
    Approximation error a = 0 (exact block encoding). main.tex:1098-1109 -
    gatesBound : String
    Gate-count bound: O(∑_g Q_g n log n + κ n). main.tex:1105-1108 -
    pureAncillas : ℕ
    Pure ancilla qubits: 2n. main.tex:1107 -
    obligations : QuantumBlockEncoding.GHL2025.RobinProofObligations
    All proof obligations for this theorem. main.tex:1098-1109 -
Definition10.2.45
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “default one term robin theorem data”. Default theorem data instance from concrete parameters.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Default theorem data instance from concrete parameters. main.tex:1098-1109 -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:700. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.45●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.defaultOneTermRobinTheoremData
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GHL2025.OneTermRobinTheoremData
    def QuantumBlockEncoding.GHL2025.defaultOneTermRobinTheoremData
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GHL2025.OneTermRobinTheoremData
    Default theorem data instance from concrete parameters. main.tex:1098-1109 -
Definition10.2.46
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “robin boundary rotation angle”. A proposition-valued field is a requirement until a constructor supplies it. A controlled R_y rotation angle for a single boundary row entry.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. A controlled R_y rotation angle for a single boundary row entry. The paper (Eq. angles for Ry, main.tex:1081-1083) defines: theta_j^s = arccos(D_j^(s) / N_D) for sparse index s in {0,...,kappa-1} and boundary row j. main.tex:1081-1083 -

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:714. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.46●1 definition
  • structure(4 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.RobinBoundaryRotationAngle : Type
    structure QuantumBlockEncoding.GHL2025.RobinBoundaryRotationAngle :
  Type
    A controlled R_y rotation angle for a single boundary row entry.
The paper (Eq. angles for Ry, main.tex:1081-1083) defines:
  theta_j^s = arccos(D_j^(s) / N_D)
for sparse index s in {0,...,kappa-1} and boundary row j.
main.tex:1081-1083 -

    Fields

    row : ℕ
    Row index j (boundary row: j < K1 or j > K2). main.tex:1082 -
    sparseIndex : ℕ
    Sparse index s in {0,...,kappa-1}. main.tex:1082 -
    matrixEntry : QuantumBlockEncoding.Coeff
    The matrix entry D_j^(s) being encoded. main.tex:1082 -
    arccosArgument : QuantumBlockEncoding.Coeff
    The argument to arccos: D_j^(s) / N_D (symbolic Coeff).
The caller must ensure this evaluates to a value in [-1, 1].
main.tex:1081-1083 -
Definition10.2.47
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “robin boundary rotation set”. A proposition-valued field is a requirement until a constructor supplies it. The set of all boundary-controlled rotation angles for a given Robin construction.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The set of all boundary-controlled rotation angles for a given Robin construction. For each boundary row j and sparse index s, there is one angle theta_j^s. Total count = kappa * (K1 + gridSize - K2) = kappa * deviatingIndices. main.tex:1081-1083, 1088-1089 -

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:732. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.47●1 definition
  • structure(6 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.RobinBoundaryRotationSet : Type
    structure QuantumBlockEncoding.GHL2025.RobinBoundaryRotationSet :
  Type
    The set of all boundary-controlled rotation angles for a given Robin construction.
For each boundary row j and sparse index s, there is one angle theta_j^s.
Total count = kappa * (K1 + gridSize - K2) = kappa * deviatingIndices.
main.tex:1081-1083, 1088-1089 -

    Fields

    K1 : ℕ
    Bulk window lower bound. main.tex:1095 -
    K2 : ℕ
    Bulk window upper bound. main.tex:1095 -
    gridSize : ℕ
    Grid size = 2^n. main.tex:1082 -
    kappa : ℕ
    Diagonal sparsity bound (number of nonzero entries per row). main.tex:1075 -
    normalizerND : QuantumBlockEncoding.Coeff
    Normalizer N_D >= ||D||_max. main.tex:1085 -
    angles : List QuantumBlockEncoding.GHL2025.RobinBoundaryRotationAngle
    Individual rotation angles, one per (boundary_row, sparse_index) pair.
main.tex:1081-1083 -
Definition10.2.48
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “expected count”. Number of boundary rows = K1 + gridSize - K2.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Number of boundary rows = K1 + gridSize - K2. Each boundary row has kappa rotation angles (one per sparse index). main.tex:1092-1095 -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:752. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.48●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.RobinBoundaryRotationSet.expectedCount
  (rs : QuantumBlockEncoding.GHL2025.RobinBoundaryRotationSet) : ℕ
    def QuantumBlockEncoding.GHL2025.RobinBoundaryRotationSet.expectedCount
  (rs :
    QuantumBlockEncoding.GHL2025.RobinBoundaryRotationSet) :
  ℕ
    Number of boundary rows = K1 + gridSize - K2.
Each boundary row has kappa rotation angles (one per sparse index).
main.tex:1092-1095 -
Definition10.2.49
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “imported claims”.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:755. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.49●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.importedClaims :
  List QuantumBlockEncoding.ConstructionClaim
    def QuantumBlockEncoding.GHL2025.importedClaims :
  List
    QuantumBlockEncoding.ConstructionClaim
Definition10.2.50
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “one term robin total qubits”. Total number of qubits in the one-term Robin circuit.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Total number of qubits in the one-term Robin circuit. Uses the register partition total: sum of all register widths. main.tex:1098-1109 -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:770. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.50●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) : ℕ
    def QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  ℕ
    Total number of qubits in the one-term Robin circuit.
Uses the register partition total: sum of all register widths.
main.tex:1098-1109 -
Definition10.2.51
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “effective robin signal qubits”. Effective signal qubits: total circuit qubits minus the system register width.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Effective signal qubits: total circuit qubits minus the system register width. This is the number of non-system qubits in the register partition. main.tex:1098-1109 -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:777. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.51●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.effectiveRobinSignalQubits
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) : ℕ
    def QuantumBlockEncoding.GHL2025.effectiveRobinSignalQubits
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  ℕ
    Effective signal qubits: total circuit qubits minus the system register width.
This is the number of non-system qubits in the register partition.
main.tex:1098-1109 -
Theorem10.2.52
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “default one term robin theorem data signal qubits eq layout”; the hypotheses and conclusion in the code panel fix its exact scope. The theorem tuple uses the paper's signal-qubit count.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The theorem tuple uses the paper's signal-qubit count. This is the block-encoding parameter 'ceil(log2 n) + ceil(log2 G_f) + ceil(log2 kappa) + 4', not the number of all non-system wires in the concrete circuit register partition. Guseynov-Huang-Liu 2025, Theorem one-term block-encoding, arXiv:2506.20478.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:789. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.52●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.defaultOneTermRobinTheoremData_signalQubits_eq_layout
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  (QuantumBlockEncoding.GHL2025.defaultOneTermRobinTheoremData
        p).signalQubits =
    (QuantumBlockEncoding.GHL2025.oneTermRobinLayout p).signalQubits
    theorem QuantumBlockEncoding.GHL2025.defaultOneTermRobinTheoremData_signalQubits_eq_layout
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  (QuantumBlockEncoding.GHL2025.defaultOneTermRobinTheoremData
        p).signalQubits =
    (QuantumBlockEncoding.GHL2025.oneTermRobinLayout
        p).signalQubits
    The theorem tuple uses the paper's signal-qubit count.

This is the block-encoding parameter
`ceil(log2 n) + ceil(log2 G_f) + ceil(log2 kappa) + 4`, not the number of
all non-system wires in the concrete circuit register partition.
Guseynov-Huang-Liu 2025, Theorem one-term block-encoding,
arXiv:2506.20478.
Theorem10.2.53
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “default one term robin theorem data pure ancillas eq layout”; the hypotheses and conclusion in the code panel fix its exact scope. The theorem tuple and the reusable layout record carry the same '2n' pure-ancilla resource count.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The theorem tuple and the reusable layout record carry the same '2n' pure-ancilla resource count.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:798. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.53●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.defaultOneTermRobinTheoremData_pureAncillas_eq_layout
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  (QuantumBlockEncoding.GHL2025.defaultOneTermRobinTheoremData
        p).pureAncillas =
    (QuantumBlockEncoding.GHL2025.oneTermRobinLayout p).pureAncillas
    theorem QuantumBlockEncoding.GHL2025.defaultOneTermRobinTheoremData_pureAncillas_eq_layout
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  (QuantumBlockEncoding.GHL2025.defaultOneTermRobinTheoremData
        p).pureAncillas =
    (QuantumBlockEncoding.GHL2025.oneTermRobinLayout
        p).pureAncillas
    The theorem tuple and the reusable layout record carry the same `2n`
pure-ancilla resource count.
Theorem10.2.54
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “default one term robin theorem data pure ancillas eq resource”; the hypotheses and conclusion in the code panel fix its exact scope. The theorem tuple and concrete resource record carry the same '2n' pure-ancilla count.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The theorem tuple and concrete resource record carry the same '2n' pure-ancilla count.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:807. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.54●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.defaultOneTermRobinTheoremData_pureAncillas_eq_resource
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  (QuantumBlockEncoding.GHL2025.defaultOneTermRobinTheoremData
        p).pureAncillas =
    (QuantumBlockEncoding.GHL2025.oneTermRobinResource p).pureAncilla
    theorem QuantumBlockEncoding.GHL2025.defaultOneTermRobinTheoremData_pureAncillas_eq_resource
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  (QuantumBlockEncoding.GHL2025.defaultOneTermRobinTheoremData
        p).pureAncillas =
    (QuantumBlockEncoding.GHL2025.oneTermRobinResource
        p).pureAncilla
    The theorem tuple and concrete resource record carry the same `2n`
pure-ancilla count.
Theorem10.2.55
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “effective robin signal qubits eq layout signal plus visible workspace”; the hypotheses and conclusion in the code panel fix its exact scope. The concrete block projection has to project all non-system wires.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The concrete block projection has to project all non-system wires. Compared with the theorem-level signal parameter, the circuit-level projection also includes the visible padded 'O_D^BS' pure-register qubits and the trailing one-qubit ancilla in the register partition. This is an arithmetic bridge between the theorem tuple and the matrix backend, not a block-correctness proof.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:821. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.55●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.effectiveRobinSignalQubits_eq_layout_signal_plus_visibleWorkspace
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GHL2025.effectiveRobinSignalQubits p =
    (QuantumBlockEncoding.GHL2025.oneTermRobinLayout p).signalQubits +
        (QuantumBlockEncoding.GHL2025.defaultRobinRegisterPartition
            p).odPureAncillaQubits +
      1
    theorem QuantumBlockEncoding.GHL2025.effectiveRobinSignalQubits_eq_layout_signal_plus_visibleWorkspace
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GHL2025.effectiveRobinSignalQubits
      p =
    (QuantumBlockEncoding.GHL2025.oneTermRobinLayout
            p).signalQubits +
        (QuantumBlockEncoding.GHL2025.defaultRobinRegisterPartition
            p).odPureAncillaQubits +
      1
    The concrete block projection has to project all non-system wires.

Compared with the theorem-level signal parameter, the circuit-level projection
also includes the visible padded `O_D^BS` pure-register qubits and the trailing
one-qubit ancilla in the register partition.  This is an arithmetic bridge
between the theorem tuple and the matrix backend, not a block-correctness
proof.
Theorem10.2.56
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “effective robin signal qubits eq theorem data signal plus visible workspace”; the hypotheses and conclusion in the code panel fix its exact scope. Same projection bridge, stated directly against the theorem-data tuple.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Same projection bridge, stated directly against the theorem-data tuple. This pins the matrix backend's projection dimension to the paper theorem's signal parameter plus the visible padded 'O_D^BS' workspace and the one-qubit ancilla. It is still only a layout bridge, not an ancilla-cleanup or block correctness proof.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:839. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.56●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.effectiveRobinSignalQubits_eq_theoremData_signal_plus_visibleWorkspace
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GHL2025.effectiveRobinSignalQubits p =
    (QuantumBlockEncoding.GHL2025.defaultOneTermRobinTheoremData
            p).signalQubits +
        (QuantumBlockEncoding.GHL2025.defaultRobinRegisterPartition
            p).odPureAncillaQubits +
      1
    theorem QuantumBlockEncoding.GHL2025.effectiveRobinSignalQubits_eq_theoremData_signal_plus_visibleWorkspace
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GHL2025.effectiveRobinSignalQubits
      p =
    (QuantumBlockEncoding.GHL2025.defaultOneTermRobinTheoremData
            p).signalQubits +
        (QuantumBlockEncoding.GHL2025.defaultRobinRegisterPartition
            p).odPureAncillaQubits +
      1
    Same projection bridge, stated directly against the theorem-data tuple.

This pins the matrix backend's projection dimension to the paper theorem's
signal parameter plus the visible padded `O_D^BS` workspace and the one-qubit
ancilla.  It is still only a layout bridge, not an ancilla-cleanup or block
correctness proof.
Definition10.2.57
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “robin indicator bit position”. Bit position of the indicator qubit in the compound register.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Bit position of the indicator qubit in the compound register. = ancillaQubit + systemQubits + odPureAncillaQubits + sparseIndexQubits = 1 + n + (n - clog2 κ) + clog2 κ = 1 + 2n main.tex:1113 -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:852. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.57●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.robinIndicatorBitPosition
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) : ℕ
    def QuantumBlockEncoding.GHL2025.robinIndicatorBitPosition
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  ℕ
    Bit position of the indicator qubit in the compound register.
= ancillaQubit + systemQubits + odPureAncillaQubits + sparseIndexQubits
= 1 + n + (n - clog2 κ) + clog2 κ = 1 + 2n
main.tex:1113 -
Definition10.2.58
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “robin sparse column map”. Column mapping for the banded sparse access oracle O_D^BS.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Column mapping for the banded sparse access oracle O_D^BS. Returns the column index for sparse index s in row i of the Robin derivative matrix. For bulk rows (K1 ≤ i ≤ K2): 5 entries, col(s,i) = i - 2 + s for s < 5. For left boundary: - Row 0 (3 entries): col(s,0) = s for s < 3 - Row 1 (4 entries): col(s,1) = s for s < 4 For right boundary (N = gridSize n): - Row N-2 (4 entries): col(s,N-2) = N-4+s for s < 4 - Row N-1 (3 entries): col(s,N-1) = N-3+s for s < 3 For unused sparse indices (s ≥ entry count): returns i (identity on system register). main.tex:784-801 -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:869. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.58●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.robinSparseColumnMap (n s i : ℕ) : ℕ
    def QuantumBlockEncoding.GHL2025.robinSparseColumnMap
  (n s i : ℕ) : ℕ
    Column mapping for the banded sparse access oracle O_D^BS.
Returns the column index for sparse index s in row i of the Robin derivative matrix.

For bulk rows (K1 ≤ i ≤ K2): 5 entries, col(s,i) = i - 2 + s for s < 5.
For left boundary:
  - Row 0 (3 entries): col(s,0) = s for s < 3
  - Row 1 (4 entries): col(s,1) = s for s < 4
For right boundary (N = gridSize n):
  - Row N-2 (4 entries): col(s,N-2) = N-4+s for s < 4
  - Row N-1 (3 entries): col(s,N-1) = N-3+s for s < 3
For unused sparse indices (s ≥ entry count): returns i (identity on system register).
main.tex:784-801 -
Definition10.2.59
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “one term robin global sparse offset”. Global sparse-slot offset table for the one-term Robin 'κ = 7' construction.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Global sparse-slot offset table for the one-term Robin 'κ = 7' construction. The first five slots keep the existing fourth-order stencil order '{-2,-1,0,1,2}'. The final two slots record the boundary-effect diagonals '{-3,3}' required by the source audit. The active Lemma 1 address uses this global slot table; row-dependent branch deletion remains only a rejected-model helper. Guseynov-Huang-Liu 2025, Lemma 'Diagonal sparsity', Lemma 'Banded-sparse-access-oracle', and Remark 'sparsity maximum', arXiv:2506.20478.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:897. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.59●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseOffset (n s : ℕ) :
  ℕ
    def QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseOffset
  (n s : ℕ) : ℕ
    Global sparse-slot offset table for the one-term Robin `κ = 7` construction.

The first five slots keep the existing fourth-order stencil order
`{-2,-1,0,1,2}`.  The final two slots record the boundary-effect diagonals
`{-3,3}` required by the source audit.  The active Lemma 1 address uses this
global slot table; row-dependent branch deletion remains only a rejected-model
helper.
Guseynov-Huang-Liu 2025, Lemma `Diagonal sparsity`, Lemma
`Banded-sparse-access-oracle`, and Remark `sparsity maximum`,
arXiv:2506.20478.
Definition10.2.60
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “one term robin global sparse address”. Global sparse-access address 'r_{si}=r_{s0}+i mod 2^n'.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Global sparse-access address 'r_{si}=r_{s0}+i mod 2^n'. This is the active paper address for 'O_D^BS'; it does not remove zero boundary-amplitude slots from the sparse register.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:915. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.60●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress
  (n s i : ℕ) : ℕ
    def QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress
  (n s i : ℕ) : ℕ
    Global sparse-access address `r_{si}=r_{s0}+i mod 2^n`.

This is the active paper address for `O_D^BS`; it does not remove zero
boundary-amplitude slots from the sparse register.
Theorem10.2.61
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin global sparse address lt grid size”; the hypotheses and conclusion in the code panel fix its exact scope. The global sparse-slot address is always an 'n'-bit row address.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The global sparse-slot address is always an 'n'-bit row address.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:919. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.61●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress_lt_gridSize
  (n s i : ℕ) :
  QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress n s i <
    QuantumBlockEncoding.gridSize n
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress_lt_gridSize
  (n s i : ℕ) :
  QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress
      n s i <
    QuantumBlockEncoding.gridSize n
    The global sparse-slot address is always an `n`-bit row address.
Definition10.2.62
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “one term robin global sparse inverse slot”. Inverse sparse slot used by the post-SWAP cleanup candidate for the global offset table.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Inverse sparse slot used by the post-SWAP cleanup candidate for the global offset table. This is only an executable preimage witness helper. It does not assert inverse uniqueness or promote the dagger-cleanup obligation.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:934. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.62●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot
  (s : ℕ) : ℕ
    def QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot
  (s : ℕ) : ℕ
    Inverse sparse slot used by the post-SWAP cleanup candidate for the global
offset table.

This is only an executable preimage witness helper.  It does not assert
inverse uniqueness or promote the dagger-cleanup obligation.
Theorem10.2.63
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin global sparse inverse slot lt eight”; the hypotheses and conclusion in the code panel fix its exact scope. The global inverse-slot helper fits in the three-bit sparse register.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The global inverse-slot helper fits in the three-bit sparse register.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:946. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.63●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot_lt_eight
  (s : ℕ) :
  QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot s < 8
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot_lt_eight
  (s : ℕ) :
  QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot
      s <
    8
    The global inverse-slot helper fits in the three-bit sparse register.
Theorem10.2.64
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin global sparse inverse slot lt seven”; the hypotheses and conclusion in the code panel fix its exact scope. The inverse sparse-slot helper stays in the active seven-slot table.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The inverse sparse-slot helper stays in the active seven-slot table.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:952. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.64●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot_lt_seven
  {s : ℕ} (hs : s < 7) :
  QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot s < 7
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot_lt_seven
  {s : ℕ} (hs : s < 7) :
  QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot
      s <
    7
    The inverse sparse-slot helper stays in the active seven-slot table.
Theorem10.2.65
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin global sparse inverse slot involutive of lt seven”; the hypotheses and conclusion in the code panel fix its exact scope. The inverse sparse-slot helper is an involution on the active 'κ = 7' slot set.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The inverse sparse-slot helper is an involution on the active 'κ = 7' slot set. This is a finite-table proof block for the global-source preimage route; it does not prove uniqueness of the full 'O_D^BS' matrix image.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:967. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.65●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot_involutive_of_lt_seven
  {s : ℕ} (hs : s < 7) :
  QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot
      (QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot
        s) =
    s
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot_involutive_of_lt_seven
  {s : ℕ} (hs : s < 7) :
  QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot
      (QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot
        s) =
    s
    The inverse sparse-slot helper is an involution on the active `κ = 7` slot
set.  This is a finite-table proof block for the global-source preimage route;
it does not prove uniqueness of the full `O_D^BS` matrix image.
Theorem10.2.66
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin global sparse inverse slot injective of lt seven”; the hypotheses and conclusion in the code panel fix its exact scope. The inverse sparse-slot helper is injective on the active 'κ = 7' slot set.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The inverse sparse-slot helper is injective on the active 'κ = 7' slot set. This feeds the later clean-preimage uniqueness proof for the corrected global slot model, but it intentionally leaves the semantic cleanup and unitarity obligations false.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:985. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.66●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot_injective_of_lt_seven
  {s t : ℕ} (hs : s < 7) (ht : t < 7)
  (h :
    QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot s =
      QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot
        t) :
  s = t
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot_injective_of_lt_seven
  {s t : ℕ} (hs : s < 7) (ht : t < 7)
  (h :
    QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot
        s =
      QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot
        t) :
  s = t
    The inverse sparse-slot helper is injective on the active `κ = 7` slot set.

This feeds the later clean-preimage uniqueness proof for the corrected global
slot model, but it intentionally leaves the semantic cleanup and unitarity
obligations false.
Theorem10.2.67
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin global sparse address inverse slot address eq”; the hypotheses and conclusion in the code panel fix its exact scope. Global sparse-address roundtrip for the supplied inverse-slot helper.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Global sparse-address roundtrip for the supplied inverse-slot helper. For an 'n >= 3' grid and any encoded sparse value below '8', addressing by a slot and then by the inverse slot returns the original row modulo '2^n'. This is the arithmetic block needed by the conditional post-SWAP preimage candidate; it is not an injectivity or cleanup proof.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:1009. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.67●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress_inverseSlot_address_eq
  {n s i : ℕ} (hn : 3 ≤ n) (hs : s < 8)
  (hi : i < QuantumBlockEncoding.gridSize n) :
  QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress n
      (QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot
        s)
      (QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress n s
        i) =
    i
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress_inverseSlot_address_eq
  {n s i : ℕ} (hn : 3 ≤ n) (hs : s < 8)
  (hi :
    i < QuantumBlockEncoding.gridSize n) :
  QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress
      n
      (QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot
        s)
      (QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress
        n s i) =
    i
    Global sparse-address roundtrip for the supplied inverse-slot helper.

For an `n >= 3` grid and any encoded sparse value below `8`, addressing by a
slot and then by the inverse slot returns the original row modulo `2^n`.
This is the arithmetic block needed by the conditional post-SWAP preimage
candidate; it is not an injectivity or cleanup proof.
Theorem10.2.68
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin global sparse offset lt grid size of lt seven”; the hypotheses and conclusion in the code panel fix its exact scope. Every active global sparse-slot offset is an 'n'-bit address when '3 ≤ n'.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Every active global sparse-slot offset is an 'n'-bit address when '3 ≤ n'.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:1131. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.68●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseOffset_lt_gridSize_of_lt_seven
  {n s : ℕ} (hn : 3 ≤ n) (hs : s < 7) :
  QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseOffset n s <
    QuantumBlockEncoding.gridSize n
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseOffset_lt_gridSize_of_lt_seven
  {n s : ℕ} (hn : 3 ≤ n) (hs : s < 7) :
  QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseOffset
      n s <
    QuantumBlockEncoding.gridSize n
    Every active global sparse-slot offset is an `n`-bit address when `3 ≤ n`.
Theorem10.2.69
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin global sparse address comp eq mod offset sum”; the hypotheses and conclusion in the code panel fix its exact scope. Composing two global sparse-slot addresses is addition by the sum of their global offsets modulo the grid size.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Composing two global sparse-slot addresses is addition by the sum of their global offsets modulo the grid size.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:1149. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.69●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress_comp_eq_mod_offset_sum
  {n s t i : ℕ} :
  QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress n t
      (QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress n s
        i) =
    ((QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseOffset n t +
            QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseOffset
              n s) %
          QuantumBlockEncoding.gridSize n +
        i) %
      QuantumBlockEncoding.gridSize n
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress_comp_eq_mod_offset_sum
  {n s t i : ℕ} :
  QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress
      n t
      (QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress
        n s i) =
    ((QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseOffset
              n t +
            QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseOffset
              n s) %
          QuantumBlockEncoding.gridSize
            n +
        i) %
      QuantumBlockEncoding.gridSize n
    Composing two global sparse-slot addresses is addition by the sum of their
global offsets modulo the grid size.
Theorem10.2.70
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin global sparse offset sum mod eq zero unique of lt seven”; the hypotheses and conclusion in the code panel fix its exact scope. If two active global sparse-slot offsets sum to zero modulo the grid, the first slot is the reverse slot of the second.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. If two active global sparse-slot offsets sum to zero modulo the grid, the first slot is the reverse slot of the second. This is only a finite arithmetic block for the corrected 'O_D^BS' address route; it does not prove dagger cleanup, unitarity, or block extraction.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:1183. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.70●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseOffset_sum_mod_eq_zero_unique_of_lt_seven
  {n s t : ℕ} (hn : 3 ≤ n) (hs : s < 7) (ht : t < 7)
  (hzero :
    (QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseOffset n t +
          QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseOffset n
            s) %
        QuantumBlockEncoding.gridSize n =
      0) :
  t = QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot s
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseOffset_sum_mod_eq_zero_unique_of_lt_seven
  {n s t : ℕ} (hn : 3 ≤ n) (hs : s < 7)
  (ht : t < 7)
  (hzero :
    (QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseOffset
            n t +
          QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseOffset
            n s) %
        QuantumBlockEncoding.gridSize n =
      0) :
  t =
    QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot
      s
    If two active global sparse-slot offsets sum to zero modulo the grid, the first
slot is the reverse slot of the second.

This is only a finite arithmetic block for the corrected `O_D^BS` address
route; it does not prove dagger cleanup, unitarity, or block extraction.
Theorem10.2.71
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin global sparse address inverse slot unique of lt seven”; the hypotheses and conclusion in the code panel fix its exact scope. Uniqueness of the reverse sparse slot for the corrected global-slot address.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Uniqueness of the reverse sparse slot for the corrected global-slot address. For active one-term Robin slots 's,t < 7', if applying slot 't' after slot 's' returns every in-range row 'i', then 't' must be the table inverse of 's'. This feeds the future clean-preimage uniqueness route while leaving all semantic 'O_D^BS' proof flags false.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:1228. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.71●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress_inverseSlot_unique_of_lt_seven
  {n s t i : ℕ} (hn : 3 ≤ n) (hs : s < 7) (ht : t < 7)
  (hi : i < QuantumBlockEncoding.gridSize n)
  (h :
    QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress n t
        (QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress n
          s i) =
      i) :
  t = QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot s
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress_inverseSlot_unique_of_lt_seven
  {n s t i : ℕ} (hn : 3 ≤ n) (hs : s < 7)
  (ht : t < 7)
  (hi :
    i < QuantumBlockEncoding.gridSize n)
  (h :
    QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress
        n t
        (QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress
          n s i) =
      i) :
  t =
    QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot
      s
    Uniqueness of the reverse sparse slot for the corrected global-slot address.

For active one-term Robin slots `s,t < 7`, if applying slot `t` after slot `s`
returns every in-range row `i`, then `t` must be the table inverse of `s`.
This feeds the future clean-preimage uniqueness route while leaving all
semantic `O_D^BS` proof flags false.
Theorem10.2.72
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin global sparse address same row injective of lt seven”; the hypotheses and conclusion in the code panel fix its exact scope. For a fixed in-range row, the corrected seven-slot global address table is injective in the sparse slot.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. For a fixed in-range row, the corrected seven-slot global address table is injective in the sparse slot. The proof reuses the reverse-slot uniqueness block rather than repeating the finite offset table. It is still only an address-level arithmetic lemma; it does not promote any 'O_D^BS' semantic flag.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:1261. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.72●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress_same_row_injective_of_lt_seven
  {n s t i : ℕ} (hn : 3 ≤ n) (hs : s < 7) (ht : t < 7)
  (hi : i < QuantumBlockEncoding.gridSize n)
  (h :
    QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress n s i =
      QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress n t
        i) :
  s = t
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress_same_row_injective_of_lt_seven
  {n s t i : ℕ} (hn : 3 ≤ n) (hs : s < 7)
  (ht : t < 7)
  (hi :
    i < QuantumBlockEncoding.gridSize n)
  (h :
    QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress
        n s i =
      QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseAddress
        n t i) :
  s = t
    For a fixed in-range row, the corrected seven-slot global address table is
injective in the sparse slot.

The proof reuses the reverse-slot uniqueness block rather than repeating the
finite offset table.  It is still only an address-level arithmetic lemma; it
does not promote any `O_D^BS` semantic flag.
Definition10.2.73
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “robin sparse column branch valid”. Row-dependent sparse-branch domain for the executable one-term Robin stencil.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Row-dependent sparse-branch domain for the executable one-term Robin stencil. This predicate is a source-contract correction candidate for Lemma 1 'O_D^BS': it marks exactly the sparse indices that correspond to nonzero stencil entries in the same five row regions used by 'robinSparseColumnMap'. The active matrix is not changed by this predicate; unused branches still remain a separate unitary-extension obligation. Guseynov-Huang-Liu 2025, Lemma 1, arXiv:2506.20478.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:1295. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.73●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.robinSparseColumnBranchValid (n s i : ℕ) :
  Bool
    def QuantumBlockEncoding.GHL2025.robinSparseColumnBranchValid
  (n s i : ℕ) : Bool
    Row-dependent sparse-branch domain for the executable one-term Robin stencil.

This predicate is a source-contract correction candidate for Lemma 1
`O_D^BS`: it marks exactly the sparse indices that correspond to nonzero
stencil entries in the same five row regions used by `robinSparseColumnMap`.
The active matrix is not changed by this predicate; unused branches still
remain a separate unitary-extension obligation.
Guseynov-Huang-Liu 2025, Lemma 1, arXiv:2506.20478.
Theorem10.2.74
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “robin sparse column branch valid boundary unused n 3”; the hypotheses and conclusion in the code panel fix its exact scope. The proposed valid-branch predicate separates the boundary unused branch that caused the recorded 'n = 3' collision, while the current executable map still sends both branches to the same address.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The proposed valid-branch predicate separates the boundary unused branch that caused the recorded 'n = 3' collision, while the current executable map still sends both branches to the same address.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:1314. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.74●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.robinSparseColumnBranchValid_boundaryUnused_n3 :
  QuantumBlockEncoding.GHL2025.robinSparseColumnBranchValid 3 0 0 =
      true ∧
    QuantumBlockEncoding.GHL2025.robinSparseColumnBranchValid 3 3 0 =
        false ∧
      QuantumBlockEncoding.GHL2025.robinSparseColumnMap 3 0 0 =
        QuantumBlockEncoding.GHL2025.robinSparseColumnMap 3 3 0
    theorem QuantumBlockEncoding.GHL2025.robinSparseColumnBranchValid_boundaryUnused_n3 :
  QuantumBlockEncoding.GHL2025.robinSparseColumnBranchValid
        3 0 0 =
      true ∧
    QuantumBlockEncoding.GHL2025.robinSparseColumnBranchValid
          3 3 0 =
        false ∧
      QuantumBlockEncoding.GHL2025.robinSparseColumnMap
          3 0 0 =
        QuantumBlockEncoding.GHL2025.robinSparseColumnMap
          3 3 0
    The proposed valid-branch predicate separates the boundary unused branch that
caused the recorded `n = 3` collision, while the current executable map still
sends both branches to the same address.
Theorem10.2.75
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “robin sparse column map lt grid size of row lt”; the hypotheses and conclusion in the code panel fix its exact scope. Proof-DAG block for the Lemma 1 address-range route.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Proof-DAG block for the Lemma 1 address-range route. For the fourth-order Robin stencil, if the input row is an 'n'-bit value and 'n >= 2', then the executable one-term column map also returns an 'n'-bit value. The paper-level contract still records 'addressRange.proved := false' because the parameter-family side condition is not yet part of 'OneTermRobinParameters'.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:1329. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.75●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.robinSparseColumnMap_lt_gridSize_of_row_lt
  {n s i : ℕ} (hn : 2 ≤ n) (hi : i < QuantumBlockEncoding.gridSize n) :
  QuantumBlockEncoding.GHL2025.robinSparseColumnMap n s i <
    QuantumBlockEncoding.gridSize n
    theorem QuantumBlockEncoding.GHL2025.robinSparseColumnMap_lt_gridSize_of_row_lt
  {n s i : ℕ} (hn : 2 ≤ n)
  (hi :
    i < QuantumBlockEncoding.gridSize n) :
  QuantumBlockEncoding.GHL2025.robinSparseColumnMap
      n s i <
    QuantumBlockEncoding.gridSize n
    Proof-DAG block for the Lemma 1 address-range route.

For the fourth-order Robin stencil, if the input row is an `n`-bit value and
`n >= 2`, then the executable one-term column map also returns an `n`-bit
value.  The paper-level contract still records `addressRange.proved := false`
because the parameter-family side condition is not yet part of
`OneTermRobinParameters`.
Definition10.2.76
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “robin sparse reverse column index”. Candidate reverse sparse index for the one-term Robin stencil.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Candidate reverse sparse index for the one-term Robin stencil. Given a target row 'target' and a post-SWAP row 'row', this returns the sparse index that would make 'row' address 'target' in the executable fourth-order Robin column map. It is only a reverse-index candidate: the checked roundtrip and cleanup obligations remain separate proof blocks. Guseynov-Huang-Liu 2025, Lemma 1, arXiv:2506.20478.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:1386. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.76●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.robinSparseReverseColumnIndex
  (n target row : ℕ) : ℕ
    def QuantumBlockEncoding.GHL2025.robinSparseReverseColumnIndex
  (n target row : ℕ) : ℕ
    Candidate reverse sparse index for the one-term Robin stencil.

Given a target row `target` and a post-SWAP row `row`, this returns the sparse
index that would make `row` address `target` in the executable fourth-order
Robin column map.  It is only a reverse-index candidate: the checked
roundtrip and cleanup obligations remain separate proof blocks.
Guseynov-Huang-Liu 2025, Lemma 1, arXiv:2506.20478.
Theorem10.2.77
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “robin sparse column map zero”; the hypotheses and conclusion in the code panel fix its exact scope. Normal form for the leftmost row of the executable Robin sparse map.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Normal form for the leftmost row of the executable Robin sparse map.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:1396. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.77●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.robinSparseColumnMap_zero (n s : ℕ) :
  QuantumBlockEncoding.GHL2025.robinSparseColumnMap n s 0 =
    if s < 3 then s else 0
    theorem QuantumBlockEncoding.GHL2025.robinSparseColumnMap_zero
  (n s : ℕ) :
  QuantumBlockEncoding.GHL2025.robinSparseColumnMap
      n s 0 =
    if s < 3 then s else 0
    Normal form for the leftmost row of the executable Robin sparse map.
Theorem10.2.78
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “robin sparse column map one”; the hypotheses and conclusion in the code panel fix its exact scope. Normal form for the second row of the executable Robin sparse map.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Normal form for the second row of the executable Robin sparse map.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:1401. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.78●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.robinSparseColumnMap_one (n s : ℕ) :
  QuantumBlockEncoding.GHL2025.robinSparseColumnMap n s 1 =
    if s < 4 then s else 1
    theorem QuantumBlockEncoding.GHL2025.robinSparseColumnMap_one
  (n s : ℕ) :
  QuantumBlockEncoding.GHL2025.robinSparseColumnMap
      n s 1 =
    if s < 4 then s else 1
    Normal form for the second row of the executable Robin sparse map.
Theorem10.2.79
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “robin sparse column map bulk”; the hypotheses and conclusion in the code panel fix its exact scope. Normal form for a bulk row of the executable Robin sparse map.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Normal form for a bulk row of the executable Robin sparse map.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:1406. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.79●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.robinSparseColumnMap_bulk (n s i : ℕ)
  (hbulk : 2 ≤ i ∧ i ≤ QuantumBlockEncoding.gridSize n - 3) :
  QuantumBlockEncoding.GHL2025.robinSparseColumnMap n s i =
    if s < 5 then i - 2 + s else i
    theorem QuantumBlockEncoding.GHL2025.robinSparseColumnMap_bulk
  (n s i : ℕ)
  (hbulk :
    2 ≤ i ∧
      i ≤
        QuantumBlockEncoding.gridSize n -
          3) :
  QuantumBlockEncoding.GHL2025.robinSparseColumnMap
      n s i =
    if s < 5 then i - 2 + s else i
    Normal form for a bulk row of the executable Robin sparse map.
Theorem10.2.80
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “robin sparse column map right boundary prev”; the hypotheses and conclusion in the code panel fix its exact scope. Normal form for the penultimate row of the executable Robin sparse map.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Normal form for the penultimate row of the executable Robin sparse map.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:1412. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.80●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.robinSparseColumnMap_rightBoundaryPrev
  {n s : ℕ} (hn : 3 ≤ n) :
  QuantumBlockEncoding.GHL2025.robinSparseColumnMap n s
      (QuantumBlockEncoding.gridSize n - 2) =
    if s < 4 then QuantumBlockEncoding.gridSize n - 4 + s
    else QuantumBlockEncoding.gridSize n - 2
    theorem QuantumBlockEncoding.GHL2025.robinSparseColumnMap_rightBoundaryPrev
  {n s : ℕ} (hn : 3 ≤ n) :
  QuantumBlockEncoding.GHL2025.robinSparseColumnMap
      n s
      (QuantumBlockEncoding.gridSize n -
        2) =
    if s < 4 then
      QuantumBlockEncoding.gridSize n -
          4 +
        s
    else
      QuantumBlockEncoding.gridSize n - 2
    Normal form for the penultimate row of the executable Robin sparse map.
Theorem10.2.81
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “robin sparse column map right boundary last”; the hypotheses and conclusion in the code panel fix its exact scope. Normal form for the last row of the executable Robin sparse map.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Normal form for the last row of the executable Robin sparse map.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:1427. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.81●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.robinSparseColumnMap_rightBoundaryLast
  {n s : ℕ} (hn : 3 ≤ n) :
  QuantumBlockEncoding.GHL2025.robinSparseColumnMap n s
      (QuantumBlockEncoding.gridSize n - 1) =
    if s < 3 then QuantumBlockEncoding.gridSize n - 3 + s
    else QuantumBlockEncoding.gridSize n - 1
    theorem QuantumBlockEncoding.GHL2025.robinSparseColumnMap_rightBoundaryLast
  {n s : ℕ} (hn : 3 ≤ n) :
  QuantumBlockEncoding.GHL2025.robinSparseColumnMap
      n s
      (QuantumBlockEncoding.gridSize n -
        1) =
    if s < 3 then
      QuantumBlockEncoding.gridSize n -
          3 +
        s
    else
      QuantumBlockEncoding.gridSize n - 1
    Normal form for the last row of the executable Robin sparse map.
Theorem10.2.82
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “robin sparse reverse column index zero”; the hypotheses and conclusion in the code panel fix its exact scope. Reverse-index normal form for row zero.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Reverse-index normal form for row zero.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:1443. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.82●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.robinSparseReverseColumnIndex_zero
  (n target : ℕ) :
  QuantumBlockEncoding.GHL2025.robinSparseReverseColumnIndex n target
      0 =
    target
    theorem QuantumBlockEncoding.GHL2025.robinSparseReverseColumnIndex_zero
  (n target : ℕ) :
  QuantumBlockEncoding.GHL2025.robinSparseReverseColumnIndex
      n target 0 =
    target
    Reverse-index normal form for row zero.
Theorem10.2.83
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “robin sparse reverse column index one”; the hypotheses and conclusion in the code panel fix its exact scope. Reverse-index normal form for row one.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Reverse-index normal form for row one.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:1448. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.83●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.robinSparseReverseColumnIndex_one
  (n target : ℕ) :
  QuantumBlockEncoding.GHL2025.robinSparseReverseColumnIndex n target
      1 =
    target
    theorem QuantumBlockEncoding.GHL2025.robinSparseReverseColumnIndex_one
  (n target : ℕ) :
  QuantumBlockEncoding.GHL2025.robinSparseReverseColumnIndex
      n target 1 =
    target
    Reverse-index normal form for row one.
Theorem10.2.84
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “robin sparse reverse column index bulk”; the hypotheses and conclusion in the code panel fix its exact scope. Reverse-index normal form for a bulk row.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Reverse-index normal form for a bulk row.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:1453. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.84●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.robinSparseReverseColumnIndex_bulk
  (n target row : ℕ)
  (hbulk : 2 ≤ row ∧ row ≤ QuantumBlockEncoding.gridSize n - 3) :
  QuantumBlockEncoding.GHL2025.robinSparseReverseColumnIndex n target
      row =
    target + 2 - row
    theorem QuantumBlockEncoding.GHL2025.robinSparseReverseColumnIndex_bulk
  (n target row : ℕ)
  (hbulk :
    2 ≤ row ∧
      row ≤
        QuantumBlockEncoding.gridSize n -
          3) :
  QuantumBlockEncoding.GHL2025.robinSparseReverseColumnIndex
      n target row =
    target + 2 - row
    Reverse-index normal form for a bulk row.
Theorem10.2.85
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “robin sparse reverse column index right boundary prev”; the hypotheses and conclusion in the code panel fix its exact scope. Reverse-index normal form for the penultimate row.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Reverse-index normal form for the penultimate row.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:1461. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.85●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.robinSparseReverseColumnIndex_rightBoundaryPrev
  {n target : ℕ} (hn : 3 ≤ n) :
  QuantumBlockEncoding.GHL2025.robinSparseReverseColumnIndex n target
      (QuantumBlockEncoding.gridSize n - 2) =
    target - (QuantumBlockEncoding.gridSize n - 4)
    theorem QuantumBlockEncoding.GHL2025.robinSparseReverseColumnIndex_rightBoundaryPrev
  {n target : ℕ} (hn : 3 ≤ n) :
  QuantumBlockEncoding.GHL2025.robinSparseReverseColumnIndex
      n target
      (QuantumBlockEncoding.gridSize n -
        2) =
    target -
      (QuantumBlockEncoding.gridSize n -
        4)
    Reverse-index normal form for the penultimate row.
Theorem10.2.86
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “robin sparse reverse column index right boundary last”; the hypotheses and conclusion in the code panel fix its exact scope. Reverse-index normal form for the last row.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Reverse-index normal form for the last row.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:1477. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.86●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.robinSparseReverseColumnIndex_rightBoundaryLast
  {n target : ℕ} (hn : 3 ≤ n) :
  QuantumBlockEncoding.GHL2025.robinSparseReverseColumnIndex n target
      (QuantumBlockEncoding.gridSize n - 1) =
    target - (QuantumBlockEncoding.gridSize n - 3)
    theorem QuantumBlockEncoding.GHL2025.robinSparseReverseColumnIndex_rightBoundaryLast
  {n target : ℕ} (hn : 3 ≤ n) :
  QuantumBlockEncoding.GHL2025.robinSparseReverseColumnIndex
      n target
      (QuantumBlockEncoding.gridSize n -
        1) =
    target -
      (QuantumBlockEncoding.gridSize n -
        3)
    Reverse-index normal form for the last row.
Theorem10.2.87
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “robin sparse reverse column roundtrip of lt eight”; the hypotheses and conclusion in the code panel fix its exact scope. The reverse sparse-index candidate is a left inverse for the executable one-term Robin column map on the three-bit sparse-index range used by the current one-term parameter family.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The reverse sparse-index candidate is a left inverse for the executable one-term Robin column map on the three-bit sparse-index range used by the current one-term parameter family. This is only the arithmetic roundtrip needed by the O_D^BS post-SWAP cleanup route. It does not prove uniqueness of the preimage, dagger cleanup, unitarity, or block correctness.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:1502. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.87●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.robinSparseReverseColumnRoundtrip_of_lt_eight
  {n s i : ℕ} (hn : 3 ≤ n) (hs : s < 8)
  (hi : i < QuantumBlockEncoding.gridSize n) :
  QuantumBlockEncoding.GHL2025.robinSparseColumnMap n
      (QuantumBlockEncoding.GHL2025.robinSparseReverseColumnIndex n i
        (QuantumBlockEncoding.GHL2025.robinSparseColumnMap n s i))
      (QuantumBlockEncoding.GHL2025.robinSparseColumnMap n s i) =
    i
    theorem QuantumBlockEncoding.GHL2025.robinSparseReverseColumnRoundtrip_of_lt_eight
  {n s i : ℕ} (hn : 3 ≤ n) (hs : s < 8)
  (hi :
    i < QuantumBlockEncoding.gridSize n) :
  QuantumBlockEncoding.GHL2025.robinSparseColumnMap
      n
      (QuantumBlockEncoding.GHL2025.robinSparseReverseColumnIndex
        n i
        (QuantumBlockEncoding.GHL2025.robinSparseColumnMap
          n s i))
      (QuantumBlockEncoding.GHL2025.robinSparseColumnMap
        n s i) =
    i
    The reverse sparse-index candidate is a left inverse for the executable
one-term Robin column map on the three-bit sparse-index range used by the
current one-term parameter family.

This is only the arithmetic roundtrip needed by the O_D^BS post-SWAP cleanup
route.  It does not prove uniqueness of the preimage, dagger cleanup, unitarity,
or block correctness.
Theorem10.2.88
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “robin sparse reverse column index lt eight of column map”; the hypotheses and conclusion in the code panel fix its exact scope. The reverse-index candidate stays inside the three-bit sparse register for columns produced by the executable one-term Robin map.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The reverse-index candidate stays inside the three-bit sparse register for columns produced by the executable one-term Robin map. This is paired with 'robinSparseReverseColumnRoundtrip_of_lt_eight'; it is a local arithmetic block for the post-SWAP preimage route, not a uniqueness or dagger-cleanup proof.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:1715. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.88●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.robinSparseReverseColumnIndex_lt_eight_of_columnMap
  {n s i : ℕ} (hn : 3 ≤ n) (hs : s < 8)
  (hi : i < QuantumBlockEncoding.gridSize n) :
  QuantumBlockEncoding.GHL2025.robinSparseReverseColumnIndex n i
      (QuantumBlockEncoding.GHL2025.robinSparseColumnMap n s i) <
    8
    theorem QuantumBlockEncoding.GHL2025.robinSparseReverseColumnIndex_lt_eight_of_columnMap
  {n s i : ℕ} (hn : 3 ≤ n) (hs : s < 8)
  (hi :
    i < QuantumBlockEncoding.gridSize n) :
  QuantumBlockEncoding.GHL2025.robinSparseReverseColumnIndex
      n i
      (QuantumBlockEncoding.GHL2025.robinSparseColumnMap
        n s i) <
    8
    The reverse-index candidate stays inside the three-bit sparse register for
columns produced by the executable one-term Robin map.

This is paired with `robinSparseReverseColumnRoundtrip_of_lt_eight`; it is a
local arithmetic block for the post-SWAP preimage route, not a uniqueness or
dagger-cleanup proof.
Definition10.2.89
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “robin sparse reverse column roundtrip check”. Executable finite audit for the reverse-index candidate.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Executable finite audit for the reverse-index candidate. For each sparse-index value below 'sparseBound' and each row of the 'n'-qubit grid, this checks that forward addressing followed by 'robinSparseReverseColumnIndex' returns to the original row. A true result is local evidence for the inverse-on-range route; it is not a semantic cleanup proof for 'O_D^BS'.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:1875. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.89●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.robinSparseReverseColumnRoundtripCheck
  (n sparseBound : ℕ) : Bool
    def QuantumBlockEncoding.GHL2025.robinSparseReverseColumnRoundtripCheck
  (n sparseBound : ℕ) : Bool
    Executable finite audit for the reverse-index candidate.

For each sparse-index value below `sparseBound` and each row of the `n`-qubit
grid, this checks that forward addressing followed by
`robinSparseReverseColumnIndex` returns to the original row.  A true result is
local evidence for the inverse-on-range route; it is not a semantic cleanup
proof for `O_D^BS`.
Definition10.2.90
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “banded sparse access paper registers”. A proposition-valued field is a requirement until a constructor supplies it. Register values used by the faithful Lemma 1 'O_D^BS' contract.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Register values used by the faithful Lemma 1 'O_D^BS' contract. The current compound-index convention stores the row register in bits '[1, 1+n)' and the paper's padded sparse-address register in bits '[1+n, 1+2n)'. Inside that address block, the low 'n - clog2 kappa' bits are the padded-zero workspace and the remaining bits encode the sparse index 's'. Guseynov-Huang-Liu 2025, Lemma 1, arXiv:2506.20478.

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:1890. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.90●1 definition
  • structure(4 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.BandedSparseAccessPaperRegisters : Type
    structure QuantumBlockEncoding.GHL2025.BandedSparseAccessPaperRegisters :
  Type
    Register values used by the faithful Lemma 1 `O_D^BS` contract.

The current compound-index convention stores the row register in bits
`[1, 1+n)` and the paper's padded sparse-address register in bits
`[1+n, 1+2n)`.  Inside that address block, the low `n - clog2 kappa` bits are
the padded-zero workspace and the remaining bits encode the sparse index `s`.
Guseynov-Huang-Liu 2025, Lemma 1, arXiv:2506.20478.

    Fields

    odRegisterValue : ℕ
    paddedZeroValue : ℕ
    sparseIndexValue : ℕ
    rowValue : ℕ
Definition10.2.91
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access paper registers”. Extract the Lemma 1 padded sparse-address and row registers from a compound basis index.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Extract the Lemma 1 padded sparse-address and row registers from a compound basis index. This is a source-contract skeleton only; it does not alter the interim 'bandedSparseAccessMatrix' helper and does not prove unitarity.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:1902. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.91●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessPaperRegisters
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessPaperRegisters
    Extract the Lemma 1 padded sparse-address and row registers from a compound
basis index.  This is a source-contract skeleton only; it does not alter the
interim `bandedSparseAccessMatrix` helper and does not prove unitarity.
Theorem10.2.92
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper registers row lt grid size”; the hypotheses and conclusion in the code panel fix its exact scope. The row field extracted for Lemma 1 is always an 'n'-bit row value.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The row field extracted for Lemma 1 is always an 'n'-bit row value.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:1919. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.92●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters_row_lt_gridSize
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
        j).rowValue <
    QuantumBlockEncoding.gridSize p.n
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters_row_lt_gridSize
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
        p j).rowValue <
    QuantumBlockEncoding.gridSize p.n
    The row field extracted for Lemma 1 is always an `n`-bit row value.
Theorem10.2.93
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper registers sparse index value eq”; the hypotheses and conclusion in the code panel fix its exact scope. The sparse-index field is the high sparse slice of the full O_D register.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The sparse-index field is the high sparse slice of the full O_D register.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:1928. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.93●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters_sparseIndexValue_eq
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
        j).sparseIndexValue =
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
            j).odRegisterValue >>>
        (p.n - QuantumBlockEncoding.clog2 p.kappa) &&&
      1 <<< QuantumBlockEncoding.clog2 p.kappa - 1
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters_sparseIndexValue_eq
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
        p j).sparseIndexValue =
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
            p j).odRegisterValue >>>
        (p.n -
          QuantumBlockEncoding.clog2
            p.kappa) &&&
      1 <<<
          QuantumBlockEncoding.clog2
            p.kappa -
        1
    The sparse-index field is the high sparse slice of the full O_D register.
Theorem10.2.94
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper registers padded zero value eq”; the hypotheses and conclusion in the code panel fix its exact scope. The padded-zero field is the low padded slice of the full O_D register.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The padded-zero field is the low padded slice of the full O_D register.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:1936. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.94●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters_paddedZeroValue_eq
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
        j).paddedZeroValue =
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
          j).odRegisterValue &&&
      1 <<< (p.n - QuantumBlockEncoding.clog2 p.kappa) - 1
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters_paddedZeroValue_eq
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
        p j).paddedZeroValue =
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
          p j).odRegisterValue &&&
      1 <<<
          (p.n -
            QuantumBlockEncoding.clog2
              p.kappa) -
        1
    The padded-zero field is the low padded slice of the full O_D register.
Theorem10.2.95
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper registers sparse index lt”; the hypotheses and conclusion in the code panel fix its exact scope. The extracted sparse-index field always fits in its declared bit width.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The extracted sparse-index field always fits in its declared bit width.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:1944. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.95●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters_sparseIndex_lt
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
        j).sparseIndexValue <
    1 <<< QuantumBlockEncoding.clog2 p.kappa
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters_sparseIndex_lt
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
        p j).sparseIndexValue <
    1 <<<
      QuantumBlockEncoding.clog2 p.kappa
    The extracted sparse-index field always fits in its declared bit width.
Theorem10.2.96
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper registers od register value lt”; the hypotheses and conclusion in the code panel fix its exact scope. The extracted O_D register value always fits in its declared 'n'-bit block.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The extracted O_D register value always fits in its declared 'n'-bit block.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:1955. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.96●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters_odRegisterValue_lt
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
        j).odRegisterValue <
    1 <<< p.n
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters_odRegisterValue_lt
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
        p j).odRegisterValue <
    1 <<< p.n
    The extracted O_D register value always fits in its declared `n`-bit block.
Definition10.2.97
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access row dependent paper address”. Rejected row-dependent paper-address helper.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Rejected row-dependent paper-address helper. This is the old active address model: it used 'robinSparseColumnMap', which deletes boundary zero-amplitude sparse slots by folding them back to the row. It is retained only for regression tests and proof-attempt memory.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:1971. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.97●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessRowDependentPaperAddress
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) : ℕ
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessRowDependentPaperAddress
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) : ℕ
    Rejected row-dependent paper-address helper.

This is the old active address model: it used `robinSparseColumnMap`, which
deletes boundary zero-amplitude sparse slots by folding them back to the row.
It is retained only for regression tests and proof-attempt memory.
Definition10.2.98
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access paper address”. Paper address value 'r_si' for the one-term Robin sparse-access oracle.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Paper address value 'r_si' for the one-term Robin sparse-access oracle. The active address follows the global sparse-slot formula 'r_si = r_s0 + i mod 2^n'. Boundary or zero-amplitude slots remain present in the sparse register; the coefficient layer supplies zero values where needed.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:1983. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.98●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) : ℕ
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) : ℕ
    Paper address value `r_si` for the one-term Robin sparse-access oracle.

The active address follows the global sparse-slot formula
`r_si = r_s0 + i mod 2^n`.  Boundary or zero-amplitude slots remain present in
the sparse register; the coefficient layer supplies zero values where needed.
Definition10.2.99
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access paper address in range”. Executable check that the paper address 'r_si' fits in the n-bit address register.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Executable check that the paper address 'r_si' fits in the n-bit address register.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:1988. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.99●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddressInRange
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  Bool
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddressInRange
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) : Bool
    Executable check that the paper address `r_si` fits in the n-bit address register.
Theorem10.2.100
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper address in range iff”; the hypotheses and conclusion in the code panel fix its exact scope. Boolean form of the executable 'O_D^BS' address-range check.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Boolean form of the executable 'O_D^BS' address-range check.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:1992. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.100●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddressInRange_iff
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddressInRange p
        j =
      true ↔
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p j <
      1 <<< p.n
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddressInRange_iff
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddressInRange
        p j =
      true ↔
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
        p j <
      1 <<< p.n
    Boolean form of the executable `O_D^BS` address-range check.
Theorem10.2.101
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper address lt grid size of two le”; the hypotheses and conclusion in the code panel fix its exact scope. The executable paper address is in range for the fourth-order grid regime '2 <= n'.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The executable paper address is in range for the fourth-order grid regime '2 <= n'. This is a reusable arithmetic block; it does not promote the paper-level semantic obligation.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2004. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.101●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress_lt_gridSize_of_two_le
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ)
  (_hn : 2 ≤ p.n) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p j <
    QuantumBlockEncoding.gridSize p.n
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress_lt_gridSize_of_two_le
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) (_hn : 2 ≤ p.n) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
      p j <
    QuantumBlockEncoding.gridSize p.n
    The executable paper address is in range for the fourth-order grid regime
`2 <= n`.  This is a reusable arithmetic block; it does not promote the
paper-level semantic obligation.
Theorem10.2.102
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper address in range eq true of two le”; the hypotheses and conclusion in the code panel fix its exact scope. The executable address-range Boolean evaluates to true for the fourth-order grid regime '2 <= n'.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The executable address-range Boolean evaluates to true for the fourth-order grid regime '2 <= n'. The contract flag remains false until the paper parameter family records this side condition.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2017. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.102●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddressInRange_eq_true_of_two_le
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ)
  (hn : 2 ≤ p.n) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddressInRange p
      j =
    true
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddressInRange_eq_true_of_two_le
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) (hn : 2 ≤ p.n) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddressInRange
      p j =
    true
    The executable address-range Boolean evaluates to true for the fourth-order
grid regime `2 <= n`.  The contract flag remains false until the paper
parameter family records this side condition.
Definition10.2.103
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access paper image”. Executable Lemma 1 image skeleton for 'O_D^BS'.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Executable Lemma 1 image skeleton for 'O_D^BS'. It preserves the row/system register and replaces the padded sparse-address register by 'r_si'. Correctness, unitarity, and dagger cleanup remain recorded in 'defaultBandedSparseAccessPaperContract p' with 'proved := false'.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:2031. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.103●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) : ℕ
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) : ℕ
    Executable Lemma 1 image skeleton for `O_D^BS`.

It preserves the row/system register and replaces the padded sparse-address
register by `r_si`.  Correctness, unitarity, and dagger cleanup remain recorded
in `defaultBandedSparseAccessPaperContract p` with `proved := false`.
Definition10.2.104
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access row dependent paper image”. Rejected row-dependent image helper corresponding to the old active address.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Rejected row-dependent image helper corresponding to the old active address. This has the same register splice as 'bandedSparseAccessPaperImage', but writes the row-dependent helper address. It is not the active paper image.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:2047. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.104●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessRowDependentPaperImage
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) : ℕ
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessRowDependentPaperImage
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) : ℕ
    Rejected row-dependent image helper corresponding to the old active address.

This has the same register splice as `bandedSparseAccessPaperImage`, but writes
the row-dependent helper address.  It is not the active paper image.
Theorem10.2.105
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper register value eq mod”; the hypotheses and conclusion in the code panel fix its exact scope. Bit-slice extraction as arithmetic division followed by an 'n'-bit remainder.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Bit-slice extraction as arithmetic division followed by an 'n'-bit remainder. This keeps later register-splice proofs in ordinary arithmetic form.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2062. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.105●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisterValue_eq_mod
  (x offset n : ℕ) :
  x >>> offset &&& 1 <<< n - 1 = x / 2 ^ offset % 2 ^ n
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisterValue_eq_mod
  (x offset n : ℕ) :
  x >>> offset &&& 1 <<< n - 1 =
    x / 2 ^ offset % 2 ^ n
    Bit-slice extraction as arithmetic division followed by an `n`-bit remainder.
This keeps later register-splice proofs in ordinary arithmetic form.
Theorem10.2.106
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper high width le total qubits”; the hypotheses and conclusion in the code panel fix its exact scope. The O_D^BS address block ends before the full one-term Robin basis width.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The O_D^BS address block ends before the full one-term Robin basis width.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2076. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.106●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperHighWidth_le_totalQubits
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  1 + 2 * p.n ≤ QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperHighWidth_le_totalQubits
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  1 + 2 * p.n ≤
    QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
      p
    The O_D^BS address block ends before the full one-term Robin basis width.
Theorem10.2.107
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper image low block lt high base of address lt”; the hypotheses and conclusion in the code panel fix its exact scope. The low block of the paper image fits below the high-tail boundary whenever the written O_D^BS address is an n-bit value.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The low block of the paper image fits below the high-tail boundary whenever the written O_D^BS address is an n-bit value.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2087. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.107●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage_lowBlock_lt_highBase_of_address_lt
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ)
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p j <
      1 <<< p.n) :
  have lowWidth := 1 + p.n;
  have highWidth := 1 + 2 * p.n;
  have lowBase := 2 ^ lowWidth;
  have highBase := 2 ^ highWidth;
  have lowPrefix := j % lowBase;
  have address :=
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p j;
  lowPrefix + address * lowBase < highBase
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage_lowBlock_lt_highBase_of_address_lt
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ)
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
        p j <
      1 <<< p.n) :
  have lowWidth := 1 + p.n;
  have highWidth := 1 + 2 * p.n;
  have lowBase := 2 ^ lowWidth;
  have highBase := 2 ^ highWidth;
  have lowPrefix := j % lowBase;
  have address :=
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
      p j;
  lowPrefix + address * lowBase < highBase
    The low block of the paper image fits below the high-tail boundary whenever the
written O_D^BS address is an n-bit value.
Theorem10.2.108
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper image mod low base”; the hypotheses and conclusion in the code panel fix its exact scope. The paper image preserves the low ancilla-and-row block modulo its width.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The paper image preserves the low ancilla-and-row block modulo its width.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2133. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.108●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage_mod_lowBase
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p j %
      2 ^ (1 + p.n) =
    j % 2 ^ (1 + p.n)
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage_mod_lowBase
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
        p j %
      2 ^ (1 + p.n) =
    j % 2 ^ (1 + p.n)
    The paper image preserves the low ancilla-and-row block modulo its width.
Theorem10.2.109
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper image div low base mod eq”; the hypotheses and conclusion in the code panel fix its exact scope. After shifting past the low block, the paper image exposes the written address modulo the n-bit O_D^BS register.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. After shifting past the low block, the paper image exposes the written address modulo the n-bit O_D^BS register.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2172. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.109●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage_div_lowBase_mod_eq
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ)
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p j <
      1 <<< p.n) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p j /
        2 ^ (1 + p.n) %
      2 ^ p.n =
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p j
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage_div_lowBase_mod_eq
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ)
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
        p j <
      1 <<< p.n) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
          p j /
        2 ^ (1 + p.n) %
      2 ^ p.n =
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
      p j
    After shifting past the low block, the paper image exposes the written address
modulo the n-bit O_D^BS register.
Theorem10.2.110
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper image lt qubit dim of address lt”; the hypotheses and conclusion in the code panel fix its exact scope. The executable paper image remains inside the full finite basis when the input column is in range and the written O_D^BS address is n-bit.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The executable paper image remains inside the full finite basis when the input column is in range and the written O_D^BS address is n-bit.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2219. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.110●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage_lt_qubitDim_of_address_lt
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ)
  (hj :
    j <
      QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p j <
      1 <<< p.n) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p j <
    QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage_lt_qubitDim_of_address_lt
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ)
  (hj :
    j <
      QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p))
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
        p j <
      1 <<< p.n) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
      p j <
    QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p)
    The executable paper image remains inside the full finite basis when the input
column is in range and the written O_D^BS address is n-bit.
Definition10.2.111
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access paper image fin”. Finite-basis index for the executable Lemma 1 'O_D^BS' paper image.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Finite-basis index for the executable Lemma 1 'O_D^BS' paper image. This constructor is available only when the source column is already in the full finite basis and the written 'O_D^BS' address is n-bit. It is a bridge from the arithmetic image function to matrix entries, not a unitarity proof.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:2268. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.111●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageFin
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p ↑j <
      1 <<< p.n) :
  Fin
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageFin
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
        p ↑j <
      1 <<< p.n) :
  Fin
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p))
    Finite-basis index for the executable Lemma 1 `O_D^BS` paper image.

This constructor is available only when the source column is already in the
full finite basis and the written `O_D^BS` address is n-bit.  It is a bridge
from the arithmetic image function to matrix entries, not a unitarity proof.
Theorem10.2.112
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper image fin val”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2276. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.112●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageFin_val
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p ↑j <
      1 <<< p.n) :
  ↑(QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageFin p j
        haddr) =
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p ↑j
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageFin_val
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
        p ↑j <
      1 <<< p.n) :
  ↑(QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageFin
        p j haddr) =
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
      p ↑j
Theorem10.2.113
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper image row value eq”; the hypotheses and conclusion in the code panel fix its exact scope. Register extraction from the paper image preserves the row register.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Register extraction from the paper image preserves the row register.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2284. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.113●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage_rowValue_eq
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p
          j)).rowValue =
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
        j).rowValue
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage_rowValue_eq
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
        p
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
          p j)).rowValue =
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
        p j).rowValue
    Register extraction from the paper image preserves the row register.
Theorem10.2.114
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper image od register value eq”; the hypotheses and conclusion in the code panel fix its exact scope. Register extraction from the paper image reports the written O_D^BS address.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Register extraction from the paper image reports the written O_D^BS address.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2303. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.114●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage_odRegisterValue_eq
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ)
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p j <
      1 <<< p.n) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p
          j)).odRegisterValue =
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p j
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage_odRegisterValue_eq
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ)
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
        p j <
      1 <<< p.n) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
        p
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
          p j)).odRegisterValue =
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
      p j
    Register extraction from the paper image reports the written O_D^BS address.
Definition10.2.115
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access paper high tail”. High signal/workspace bits above the n-bit 'O_D^BS' address register.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. High signal/workspace bits above the n-bit 'O_D^BS' address register.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:2315. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.115●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperHighTail
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) : ℕ
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperHighTail
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) : ℕ
    High signal/workspace bits above the n-bit `O_D^BS` address register.
Theorem10.2.116
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper image high tail eq of address lt”; the hypotheses and conclusion in the code panel fix its exact scope. The arithmetic register-splice form of 'bandedSparseAccessPaperImage' preserves all bits above the 'O_D^BS' address register when the written address is n-bit.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The arithmetic register-splice form of 'bandedSparseAccessPaperImage' preserves all bits above the 'O_D^BS' address register when the written address is n-bit. This is a proof-DAG block for Lemma 1 register safety. It does not promote the paper-level 'noSpill' obligation because the parameter-family side conditions are still tracked by 'defaultBandedSparseAccessPaperContract'.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2326. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.116●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage_highTail_eq_of_address_lt
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ)
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p j <
      1 <<< p.n) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperHighTail p
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p j) =
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperHighTail p j
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage_highTail_eq_of_address_lt
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ)
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
        p j <
      1 <<< p.n) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperHighTail
      p
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
        p j) =
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperHighTail
      p j
    The arithmetic register-splice form of `bandedSparseAccessPaperImage` preserves
all bits above the `O_D^BS` address register when the written address is n-bit.

This is a proof-DAG block for Lemma 1 register safety.  It does not promote the
paper-level `noSpill` obligation because the parameter-family side conditions
are still tracked by `defaultBandedSparseAccessPaperContract`.
Definition10.2.117
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access paper image no spill”. Executable check that the paper-image skeleton does not write past the n-bit 'O_D^BS' address register into the indicator or 'm_f' bits above it.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Executable check that the paper-image skeleton does not write past the n-bit 'O_D^BS' address register into the indicator or 'm_f' bits above it.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:2361. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.117●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageNoSpill
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  Bool
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageNoSpill
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) : Bool
    Executable check that the paper-image skeleton does not write past the n-bit
`O_D^BS` address register into the indicator or `m_f` bits above it.
Theorem10.2.118
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper image no spill iff”; the hypotheses and conclusion in the code panel fix its exact scope. Boolean form of the executable high-tail no-spill check.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Boolean form of the executable high-tail no-spill check. The high-tail theorem above discharges this Boolean under an n-bit written address, while the paper-level semantic obligation remains a separate flag.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2371. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.118●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageNoSpill_iff
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageNoSpill p j =
      true ↔
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperHighTail p
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p
          j) =
      QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperHighTail p j
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageNoSpill_iff
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageNoSpill
        p j =
      true ↔
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperHighTail
        p
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
          p j) =
      QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperHighTail
        p j
    Boolean form of the executable high-tail no-spill check.

The high-tail theorem above discharges this Boolean under an n-bit written
address, while the paper-level semantic obligation remains a separate flag.
Theorem10.2.119
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper image no spill eq true of address lt”; the hypotheses and conclusion in the code panel fix its exact scope. The no-spill Boolean follows from the executable n-bit address bound.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The no-spill Boolean follows from the executable n-bit address bound.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2380. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.119●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageNoSpill_eq_true_of_address_lt
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ)
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p j <
      1 <<< p.n) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageNoSpill p j =
    true
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageNoSpill_eq_true_of_address_lt
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ)
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
        p j <
      1 <<< p.n) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageNoSpill
      p j =
    true
    The no-spill Boolean follows from the executable n-bit address bound.
Theorem10.2.120
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper image no spill eq true of two le”; the hypotheses and conclusion in the code panel fix its exact scope. The no-spill Boolean is true in the fourth-order grid regime '2 <= n', reusing the address-range proof-DAG block.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The no-spill Boolean is true in the fourth-order grid regime '2 <= n', reusing the address-range proof-DAG block. Semantic obligation flags remain false.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2391. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.120●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageNoSpill_eq_true_of_two_le
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ)
  (hn : 2 ≤ p.n) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageNoSpill p j =
    true
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageNoSpill_eq_true_of_two_le
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) (hn : 2 ≤ p.n) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageNoSpill
      p j =
    true
    The no-spill Boolean is true in the fourth-order grid regime `2 <= n`, reusing
the address-range proof-DAG block.  Semantic obligation flags remain false.
Definition10.2.121
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access paper clean input”. Clean-domain predicate for the Lemma 1 'O_D^BS' source equation.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Clean-domain predicate for the Lemma 1 'O_D^BS' source equation. The paper specifies columns whose padded zero register is '|0>^(n-l)'. Columns outside this domain still need a separate unitary-completion proof; the current paper-image matrix is only a Phase 1 skeleton for that full-space extension.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:2406. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.121●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  Bool
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) : Bool
    Clean-domain predicate for the Lemma 1 `O_D^BS` source equation.

The paper specifies columns whose padded zero register is `|0>^(n-l)`.
Columns outside this domain still need a separate unitary-completion proof;
the current paper-image matrix is only a Phase 1 skeleton for that full-space
extension.
Definition10.2.122
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access paper sparse index in kappa”. Faithful sparse-slot range for the Lemma 1 'O_D^BS' source equation.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Faithful sparse-slot range for the Lemma 1 'O_D^BS' source equation. The paper source domain keeps the global slot 's' whenever 's < kappa'. Whether a boundary coefficient is zero is handled by the amplitude layer, not by deleting the sparse-register slot from the index oracle.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:2416. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.122●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSparseIndexInKappa
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  Bool
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSparseIndexInKappa
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) : Bool
    Faithful sparse-slot range for the Lemma 1 `O_D^BS` source equation.

The paper source domain keeps the global slot `s` whenever `s < kappa`.
Whether a boundary coefficient is zero is handled by the amplitude layer, not
by deleting the sparse-register slot from the index oracle.
Definition10.2.123
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access paper global slot source”. Faithful clean source domain for the active global-slot 'O_D^BS' address.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Faithful clean source domain for the active global-slot 'O_D^BS' address. This predicate is the padded clean input from Lemma 1 together with the global slot range 's < kappa'. It supersedes the row-dependent nonzero-branch classifier as the active source-domain contract for 'bandedSparseAccessPaperImage'.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:2427. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.123●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  Bool
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) : Bool
    Faithful clean source domain for the active global-slot `O_D^BS` address.

This predicate is the padded clean input from Lemma 1 together with the global
slot range `s < kappa`.  It supersedes the row-dependent nonzero-branch
classifier as the active source-domain contract for `bandedSparseAccessPaperImage`.
Theorem10.2.124
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper global slot source clean input eq true”; the hypotheses and conclusion in the code panel fix its exact scope. A faithful global-slot source column is clean in the padded O_D register.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. A faithful global-slot source column is clean in the padded O_D register.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2433. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.124●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource_cleanInput_eq_true
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ)
  (h :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p j =
      true) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput p j =
    true
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource_cleanInput_eq_true
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ)
  (h :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p j =
      true) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput
      p j =
    true
    A faithful global-slot source column is clean in the padded O_D register.
Theorem10.2.125
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper global slot source sparse index lt kappa”; the hypotheses and conclusion in the code panel fix its exact scope. A faithful global-slot source column has sparse index below 'kappa'.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. A faithful global-slot source column has sparse index below 'kappa'.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2442. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.125●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource_sparseIndex_lt_kappa
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ)
  (h :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p j =
      true) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
        j).sparseIndexValue <
    p.kappa
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource_sparseIndex_lt_kappa
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ)
  (h :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p j =
      true) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
        p j).sparseIndexValue <
    p.kappa
    A faithful global-slot source column has sparse index below `kappa`.
Theorem10.2.126
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper global slot source inverse slot injective”; the hypotheses and conclusion in the code panel fix its exact scope. Global-source wrapper for inverse-slot injectivity.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Global-source wrapper for inverse-slot injectivity. For the one-term Robin 'κ = 7' family, two active global-source columns with the same reverse sparse slot have the same extracted sparse slot. This is a local source-domain block for the later unique-preimage proof; it does not promote inverse-on-range, cleanup, or unitarity obligations.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2461. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.126●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource_inverseSlot_injective
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) {j₁ j₂ : ℕ}
  (hkappa : p.kappa = 7)
  (hsource₁ :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p j₁ =
      true)
  (hsource₂ :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p j₂ =
      true)
  (h :
    QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
            j₁).sparseIndexValue =
      QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
            j₂).sparseIndexValue) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
        j₁).sparseIndexValue =
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
        j₂).sparseIndexValue
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource_inverseSlot_injective
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  {j₁ j₂ : ℕ} (hkappa : p.kappa = 7)
  (hsource₁ :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p j₁ =
      true)
  (hsource₂ :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p j₂ =
      true)
  (h :
    QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
            p j₁).sparseIndexValue =
      QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
            p j₂).sparseIndexValue) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
        p j₁).sparseIndexValue =
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
        p j₂).sparseIndexValue
    Global-source wrapper for inverse-slot injectivity.

For the one-term Robin `κ = 7` family, two active global-source columns with
the same reverse sparse slot have the same extracted sparse slot.  This is a
local source-domain block for the later unique-preimage proof; it does not
promote inverse-on-range, cleanup, or unitarity obligations.
Theorem10.2.127
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper address same row injective of global slot source”; the hypotheses and conclusion in the code panel fix its exact scope. Same-row injectivity of the active paper address on the global-slot source domain.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Same-row injectivity of the active paper address on the global-slot source domain. This lifts the seven-slot address lemma through the Lemma 1 register extractor: if two active global-source columns have the same row and the same corrected paper address, then their sparse slots are equal. It does not assert matrix unitarity or dagger cleanup.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2493. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.127●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress_same_row_injective_of_globalSlotSource
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) {j₁ j₂ : ℕ}
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hsource₁ :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p j₁ =
      true)
  (hsource₂ :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p j₂ =
      true)
  (hrow :
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
          j₁).rowValue =
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
          j₂).rowValue)
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p j₁ =
      QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p
        j₂) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
        j₁).sparseIndexValue =
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
        j₂).sparseIndexValue
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress_same_row_injective_of_globalSlotSource
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  {j₁ j₂ : ℕ} (hn : 3 ≤ p.n)
  (hkappa : p.kappa = 7)
  (hsource₁ :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p j₁ =
      true)
  (hsource₂ :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p j₂ =
      true)
  (hrow :
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
          p j₁).rowValue =
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
          p j₂).rowValue)
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
        p j₁ =
      QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
        p j₂) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
        p j₁).sparseIndexValue =
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
        p j₂).sparseIndexValue
    Same-row injectivity of the active paper address on the global-slot source
domain.

This lifts the seven-slot address lemma through the Lemma 1 register extractor:
if two active global-source columns have the same row and the same corrected
paper address, then their sparse slots are equal.  It does not assert matrix
unitarity or dagger cleanup.
Definition10.2.128
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access paper valid sparse branch”. Candidate row-dependent sparse-branch domain for a basis column of Lemma 1.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Candidate row-dependent sparse-branch domain for a basis column of Lemma 1. This is deliberately separate from 'bandedSparseAccessPaperCleanInput'. The paper clean-input condition only checks the padded zero register, while this candidate also excludes row-boundary sparse indices that do not correspond to nonzero stencil entries. It is now a rejected-model audit helper; the active paper source domain is 'bandedSparseAccessPaperGlobalSlotSource'.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:2541. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.128●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidSparseBranch
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  Bool
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidSparseBranch
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) : Bool
    Candidate row-dependent sparse-branch domain for a basis column of Lemma 1.

This is deliberately separate from `bandedSparseAccessPaperCleanInput`.  The
paper clean-input condition only checks the padded zero register, while this
candidate also excludes row-boundary sparse indices that do not correspond to
nonzero stencil entries.  It is now a rejected-model audit helper; the active
paper source domain is `bandedSparseAccessPaperGlobalSlotSource`.
Definition10.2.129
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access paper valid clean source”. Candidate corrected clean source domain for Lemma 1: padded-zero input plus a row-dependent valid sparse branch.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Candidate corrected clean source domain for Lemma 1: padded-zero input plus a row-dependent valid sparse branch. This is a rejected-model contract-audit predicate only. Use 'bandedSparseAccessPaperGlobalSlotSource' for the active global-slot source contract.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:2552. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.129●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  Bool
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) : Bool
    Candidate corrected clean source domain for Lemma 1: padded-zero input plus a
row-dependent valid sparse branch.  This is a rejected-model contract-audit
predicate only.  Use `bandedSparseAccessPaperGlobalSlotSource` for the active
global-slot source contract.
Theorem10.2.130
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper valid clean source clean input eq true”; the hypotheses and conclusion in the code panel fix its exact scope. The corrected source-domain candidate implies the original clean input.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The corrected source-domain candidate implies the original clean input.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2558. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.130●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource_cleanInput_eq_true
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ)
  (h :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource
        p j =
      true) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput p j =
    true
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource_cleanInput_eq_true
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ)
  (h :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource
        p j =
      true) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput
      p j =
    true
    The corrected source-domain candidate implies the original clean input.
Theorem10.2.131
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper valid clean source valid sparse branch eq true”; the hypotheses and conclusion in the code panel fix its exact scope. The corrected source-domain candidate implies a valid sparse branch.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The corrected source-domain candidate implies a valid sparse branch.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2567. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.131●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource_validSparseBranch_eq_true
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ)
  (h :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource
        p j =
      true) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidSparseBranch
      p j =
    true
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource_validSparseBranch_eq_true
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ)
  (h :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource
        p j =
      true) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidSparseBranch
      p j =
    true
    The corrected source-domain candidate implies a valid sparse branch.
Theorem10.2.132
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper valid clean source separates boundary collision n 3”; the hypotheses and conclusion in the code panel fix its exact scope. The row-dependent valid-source audit excludes the concrete unused sparse branch from the recorded 'n = 3', 'kappa = 7' rejected-model collision.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The row-dependent valid-source audit excludes the concrete unused sparse branch from the recorded 'n = 3', 'kappa = 7' rejected-model collision. The active global-slot paper image now separates the same two clean columns.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2580. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.132●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource_separates_boundaryCollision_n3 :
  have p :=
    { n := 3, kappa := 7, functionPieces := 1,
      polynomialDegreeCost := 1 };
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput p 0 =
      true ∧
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput p
          48 =
        true ∧
      QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidSparseBranch
            p 0 =
          true ∧
        QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidSparseBranch
              p 48 =
            false ∧
          QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource
                p 0 =
              true ∧
            QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource
                  p 48 =
                false ∧
              QuantumBlockEncoding.GHL2025.bandedSparseAccessRowDependentPaperImage
                    p 0 =
                  QuantumBlockEncoding.GHL2025.bandedSparseAccessRowDependentPaperImage
                    p 48 ∧
                QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
                    p 0 ≠
                  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
                    p 48
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource_separates_boundaryCollision_n3 :
  have p :=
    { n := 3, kappa := 7,
      functionPieces := 1,
      polynomialDegreeCost := 1 };
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput
        p 0 =
      true ∧
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput
          p 48 =
        true ∧
      QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidSparseBranch
            p 0 =
          true ∧
        QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidSparseBranch
              p 48 =
            false ∧
          QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource
                p 0 =
              true ∧
            QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource
                  p 48 =
                false ∧
              QuantumBlockEncoding.GHL2025.bandedSparseAccessRowDependentPaperImage
                    p 0 =
                  QuantumBlockEncoding.GHL2025.bandedSparseAccessRowDependentPaperImage
                    p 48 ∧
                QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
                    p 0 ≠
                  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
                    p 48
    The row-dependent valid-source audit excludes the concrete unused sparse
branch from the recorded `n = 3`, `kappa = 7` rejected-model collision.  The
active global-slot paper image now separates the same two clean columns.
Definition10.2.133
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access paper unused sparse branch”. Classifier for clean padded-register columns whose sparse branch is invalid for the row-dependent Robin stencil.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Classifier for clean padded-register columns whose sparse branch is invalid for the row-dependent Robin stencil. This is the source-domain side of the unused-branch extension obligation. The active O_D^BS matrices are not changed by this predicate.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:2601. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.133●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperUnusedSparseBranch
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  Bool
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperUnusedSparseBranch
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) : Bool
    Classifier for clean padded-register columns whose sparse branch is invalid
for the row-dependent Robin stencil.

This is the source-domain side of the unused-branch extension obligation.  The
active O_D^BS matrices are not changed by this predicate.
Theorem10.2.134
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper unused sparse branch clean input eq true”; the hypotheses and conclusion in the code panel fix its exact scope. An unused sparse branch is still in the padded clean-input domain.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. An unused sparse branch is still in the padded clean-input domain.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2607. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.134●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperUnusedSparseBranch_cleanInput_eq_true
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ)
  (h :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperUnusedSparseBranch
        p j =
      true) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput p j =
    true
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperUnusedSparseBranch_cleanInput_eq_true
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ)
  (h :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperUnusedSparseBranch
        p j =
      true) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput
      p j =
    true
    An unused sparse branch is still in the padded clean-input domain.
Theorem10.2.135
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper unused sparse branch valid sparse branch eq false”; the hypotheses and conclusion in the code panel fix its exact scope. An unused sparse branch is outside the row-dependent valid-branch classifier.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. An unused sparse branch is outside the row-dependent valid-branch classifier.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2616. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.135●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperUnusedSparseBranch_validSparseBranch_eq_false
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ)
  (h :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperUnusedSparseBranch
        p j =
      true) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidSparseBranch
      p j =
    false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperUnusedSparseBranch_validSparseBranch_eq_false
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ)
  (h :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperUnusedSparseBranch
        p j =
      true) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidSparseBranch
      p j =
    false
    An unused sparse branch is outside the row-dependent valid-branch classifier.
Theorem10.2.136
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper clean domain split iff”; the hypotheses and conclusion in the code panel fix its exact scope. The executable clean padded-input domain splits into valid sparse branches and clean unused sparse branches.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The executable clean padded-input domain splits into valid sparse branches and clean unused sparse branches. This is only the local Boolean classifier split for the source-contract audit. It does not choose an image for unused branches and does not promote the semantic 'cleanDomainSplit' obligation in the full-domain wrapper.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2632. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.136●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanDomainSplit_iff
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput p j =
      true ↔
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource
          p j =
        true ∨
      QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperUnusedSparseBranch
          p j =
        true
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanDomainSplit_iff
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput
        p j =
      true ↔
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource
          p j =
        true ∨
      QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperUnusedSparseBranch
          p j =
        true
    The executable clean padded-input domain splits into valid sparse branches and
clean unused sparse branches.

This is only the local Boolean classifier split for the source-contract audit.
It does not choose an image for unused branches and does not promote the
semantic `cleanDomainSplit` obligation in the full-domain wrapper.
Theorem10.2.137
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper clean domain split disjoint”; the hypotheses and conclusion in the code panel fix its exact scope. The two branches in 'bandedSparseAccessPaperCleanDomainSplit_iff' are disjoint.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The two branches in 'bandedSparseAccessPaperCleanDomainSplit_iff' are disjoint. This is a classifier fact only; injectivity and unitary extension for the eventual image rule remain separate false obligations.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2657. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.137●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanDomainSplit_disjoint
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  ¬(QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource
          p j =
        true ∧
      QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperUnusedSparseBranch
          p j =
        true)
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanDomainSplit_disjoint
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  ¬(QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource
          p j =
        true ∧
      QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperUnusedSparseBranch
          p j =
        true)
    The two branches in `bandedSparseAccessPaperCleanDomainSplit_iff` are disjoint.

This is a classifier fact only; injectivity and unitary extension for the
eventual image rule remain separate false obligations.
Definition10.2.138
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “banded sparse access unused branch image rule contract”. A proposition-valued field is a requirement until a constructor supplies it. Interface for the missing reversible image rule on clean unused sparse branches.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Interface for the missing reversible image rule on clean unused sparse branches. No paper-backed formula has been selected yet, so 'proposedImageIndex' is 'none' and every semantic claim remains an explicit false obligation. The active 'bandedSparseAccessPaperImage' skeleton is not changed by this record.

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:2680. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.138●1 definition
  • structure(12 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.BandedSparseAccessUnusedBranchImageRuleContract :
  Type
    structure QuantumBlockEncoding.GHL2025.BandedSparseAccessUnusedBranchImageRuleContract :
  Type
    Interface for the missing reversible image rule on clean unused sparse branches.

No paper-backed formula has been selected yet, so `proposedImageIndex` is
`none` and every semantic claim remains an explicit false obligation.  The
active `bandedSparseAccessPaperImage` skeleton is not changed by this record.

    Fields

    sourceAnchor : String
    sourceIndex : ℕ
    inputRegisters : QuantumBlockEncoding.GHL2025.BandedSparseAccessPaperRegisters
    activeImageIndex : ℕ
    proposedImageIndex : Option ℕ
    cleanInput : Bool
    validSparseBranch : Bool
    unusedSparseBranch : Bool
    imageSpecified : QuantumBlockEncoding.GHL2025.ObligationRecord
    imageFinite : QuantumBlockEncoding.GHL2025.ObligationRecord
    separatesActiveCollision : QuantumBlockEncoding.GHL2025.ObligationRecord
    validBranchAgreement : QuantumBlockEncoding.GHL2025.ObligationRecord
Definition10.2.139
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access unused branch image rule contract”. Default image-rule interface for one unused-branch source column.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Default image-rule interface for one unused-branch source column. The missing reversible image is intentionally represented by 'none'; later faithful work must replace this with a paper-compatible extension before any injectivity, dagger-cleanup, or unitarity proof is attempted.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:2702. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.139●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessUnusedBranchImageRuleContract
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessUnusedBranchImageRuleContract
    Default image-rule interface for one unused-branch source column.

The missing reversible image is intentionally represented by `none`; later
faithful work must replace this with a paper-compatible extension before any
injectivity, dagger-cleanup, or unitarity proof is attempted.
Theorem10.2.140
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access unused branch image rule contract flags false”; the hypotheses and conclusion in the code panel fix its exact scope. The unused-branch image-rule interface is obligation-only in Phase 1.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The unused-branch image-rule interface is obligation-only in Phase 1.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2735. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.140●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract_flags_false
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
          p j).proposedImageIndex =
      none ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
              p j).imageSpecified.proved =
        false ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
                p j).imageFinite.proved =
          false ∧
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
                  p j).separatesActiveCollision.proved =
            false ∧
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
                  p j).validBranchAgreement.proved =
            false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract_flags_false
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
          p j).proposedImageIndex =
      none ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
              p j).imageSpecified.proved =
        false ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
                p j).imageFinite.proved =
          false ∧
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
                  p
                  j).separatesActiveCollision.proved =
            false ∧
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
                  p
                  j).validBranchAgreement.proved =
            false
    The unused-branch image-rule interface is obligation-only in Phase 1.
Theorem10.2.141
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access unused branch image rule contract of unused branch”; the hypotheses and conclusion in the code panel fix its exact scope. Classifier bridge for the unused-branch image-rule interface.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Classifier bridge for the unused-branch image-rule interface. For a clean invalid sparse branch, Lean records the branch classification and keeps the image-rule target unspecified with false proof fields.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2750. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.141●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract_of_unusedBranch
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ)
  (h :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperUnusedSparseBranch
        p j =
      true) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
          p j).cleanInput =
      true ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
            p j).validSparseBranch =
        false ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
              p j).unusedSparseBranch =
          true ∧
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
                p j).proposedImageIndex =
            none ∧
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
                    p j).imageSpecified.proved =
              false ∧
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
                    p j).validBranchAgreement.proved =
              false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract_of_unusedBranch
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ)
  (h :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperUnusedSparseBranch
        p j =
      true) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
          p j).cleanInput =
      true ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
            p j).validSparseBranch =
        false ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
              p j).unusedSparseBranch =
          true ∧
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
                p j).proposedImageIndex =
            none ∧
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
                    p
                    j).imageSpecified.proved =
              false ∧
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
                    p
                    j).validBranchAgreement.proved =
              false
    Classifier bridge for the unused-branch image-rule interface.

For a clean invalid sparse branch, Lean records the branch classification and
keeps the image-rule target unspecified with false proof fields.
Definition10.2.142
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “banded sparse access unused branch extension contract”. A proposition-valued field is a requirement until a constructor supplies it. Contract slot for a faithful reversible extension on unused sparse branches.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Contract slot for a faithful reversible extension on unused sparse branches. GHL2025 keeps zero-amplitude sparse branches inside the kappa-wide register. The current active image skeleton can collide on such branches, so Phase 1 records the missing extension as obligations instead of proving injectivity or unitarity for the colliding skeleton.

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:2775. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.142●1 definition
  • structure(13 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.BandedSparseAccessUnusedBranchExtensionContract :
  Type
    structure QuantumBlockEncoding.GHL2025.BandedSparseAccessUnusedBranchExtensionContract :
  Type
    Contract slot for a faithful reversible extension on unused sparse branches.

GHL2025 keeps zero-amplitude sparse branches inside the kappa-wide register.
The current active image skeleton can collide on such branches, so Phase 1
records the missing extension as obligations instead of proving injectivity or
unitarity for the colliding skeleton.

    Fields

    sourceAnchor : String
    inputRegisters : QuantumBlockEncoding.GHL2025.BandedSparseAccessPaperRegisters
    activeImageIndex : ℕ
    cleanInput : Bool
    validSparseBranch : Bool
    unusedSparseBranch : Bool
    unusedBranchImageRuleContract : QuantumBlockEncoding.GHL2025.BandedSparseAccessUnusedBranchImageRuleContract
    paperAgreementOnValidBranches : QuantumBlockEncoding.GHL2025.ObligationRecord
    unusedBranchImageRule : QuantumBlockEncoding.GHL2025.ObligationRecord
    unusedBranchInjective : QuantumBlockEncoding.GHL2025.ObligationRecord
    fullCleanDomainInjective : QuantumBlockEncoding.GHL2025.ObligationRecord
    daggerCleanup : QuantumBlockEncoding.GHL2025.ObligationRecord
    unitaryExtension : QuantumBlockEncoding.GHL2025.ObligationRecord
Definition10.2.143
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access unused branch extension contract”. Default unused-branch extension contract for one O_D^BS basis column.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Default unused-branch extension contract for one O_D^BS basis column. All semantic fields remain false. The record exists so later work can state the reversible completion separately from the paper image on valid branches.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:2797. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.143●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessUnusedBranchExtensionContract
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessUnusedBranchExtensionContract
    Default unused-branch extension contract for one O_D^BS basis column.

All semantic fields remain false.  The record exists so later work can state
the reversible completion separately from the paper image on valid branches.
Theorem10.2.144
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access unused branch extension contract flags false”; the hypotheses and conclusion in the code panel fix its exact scope. The unused-branch extension contract is obligation-only in Phase 1.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The unused-branch extension contract is obligation-only in Phase 1.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2837. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.144●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract_flags_false
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
            p j).paperAgreementOnValidBranches.proved =
      false ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
              p j).unusedBranchImageRule.proved =
        false ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
                p j).unusedBranchInjective.proved =
          false ∧
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
                  p j).fullCleanDomainInjective.proved =
            false ∧
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
                    p j).daggerCleanup.proved =
              false ∧
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
                    p j).unitaryExtension.proved =
              false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract_flags_false
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
            p
            j).paperAgreementOnValidBranches.proved =
      false ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
              p
              j).unusedBranchImageRule.proved =
        false ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
                p
                j).unusedBranchInjective.proved =
          false ∧
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
                  p
                  j).fullCleanDomainInjective.proved =
            false ∧
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
                    p
                    j).daggerCleanup.proved =
              false ∧
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
                    p
                    j).unitaryExtension.proved =
              false
    The unused-branch extension contract is obligation-only in Phase 1.
Theorem10.2.145
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access unused branch extension contract boundary collision n 3”; the hypotheses and conclusion in the code panel fix its exact scope. The unused-branch contract classifies the recorded row-dependent boundary collision without promoting any O_D^BS semantic proof flag.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The unused-branch contract classifies the recorded row-dependent boundary collision without promoting any O_D^BS semantic proof flag. The active global-slot image no longer has this concrete collision; the extension fields remain false because this packet does not prove full clean-domain injectivity, dagger cleanup, or unitarity.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2856. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.145●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract_boundaryCollision_n3 :
  have p :=
    { n := 3, kappa := 7, functionPieces := 1,
      polynomialDegreeCost := 1 };
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource p
        0 =
      true ∧
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperUnusedSparseBranch
          p 48 =
        true ∧
      QuantumBlockEncoding.GHL2025.bandedSparseAccessRowDependentPaperImage
            p 0 =
          QuantumBlockEncoding.GHL2025.bandedSparseAccessRowDependentPaperImage
            p 48 ∧
        QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p 0 ≠
            QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p
              48 ∧
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
                    p 48).unusedBranchInjective.proved =
              false ∧
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
                    p 48).unitaryExtension.proved =
              false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract_boundaryCollision_n3 :
  have p :=
    { n := 3, kappa := 7,
      functionPieces := 1,
      polynomialDegreeCost := 1 };
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource
        p 0 =
      true ∧
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperUnusedSparseBranch
          p 48 =
        true ∧
      QuantumBlockEncoding.GHL2025.bandedSparseAccessRowDependentPaperImage
            p 0 =
          QuantumBlockEncoding.GHL2025.bandedSparseAccessRowDependentPaperImage
            p 48 ∧
        QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
              p 0 ≠
            QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
              p 48 ∧
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
                    p
                    48).unusedBranchInjective.proved =
              false ∧
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
                    p
                    48).unitaryExtension.proved =
              false
    The unused-branch contract classifies the recorded row-dependent boundary
collision without promoting any O_D^BS semantic proof flag.

The active global-slot image no longer has this concrete collision; the
extension fields remain false because this packet does not prove full
clean-domain injectivity, dagger cleanup, or unitarity.
Theorem10.2.146
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access unused branch extension contract of unused branch”; the hypotheses and conclusion in the code panel fix its exact scope. Package the unused-branch classifier with the reversible-extension obligations.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Package the unused-branch classifier with the reversible-extension obligations. This is a contract bridge only: it exposes that an unused clean branch is in the clean padded-input domain, is outside the row-dependent valid sparse-branch classifier, and still has only false extension proof fields.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2875. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.146●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract_of_unusedBranch
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ)
  (h :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperUnusedSparseBranch
        p j =
      true) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
          p j).cleanInput =
      true ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
            p j).validSparseBranch =
        false ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
              p j).unusedSparseBranch =
          true ∧
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
                  p j).unusedBranchImageRule.proved =
            false ∧
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
                    p j).unusedBranchInjective.proved =
              false ∧
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
                      p j).fullCleanDomainInjective.proved =
                false ∧
              (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
                        p j).daggerCleanup.proved =
                  false ∧
                (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
                        p j).unitaryExtension.proved =
                  false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract_of_unusedBranch
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ)
  (h :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperUnusedSparseBranch
        p j =
      true) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
          p j).cleanInput =
      true ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
            p j).validSparseBranch =
        false ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
              p j).unusedSparseBranch =
          true ∧
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
                  p
                  j).unusedBranchImageRule.proved =
            false ∧
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
                    p
                    j).unusedBranchInjective.proved =
              false ∧
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
                      p
                      j).fullCleanDomainInjective.proved =
                false ∧
              (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
                        p
                        j).daggerCleanup.proved =
                  false ∧
                (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchExtensionContract
                        p
                        j).unitaryExtension.proved =
                  false
    Package the unused-branch classifier with the reversible-extension obligations.

This is a contract bridge only: it exposes that an unused clean branch is in
the clean padded-input domain, is outside the row-dependent valid sparse-branch
classifier, and still has only false extension proof fields.
Definition10.2.147
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “banded sparse access full clean domain extension contract”. A proposition-valued field is a requirement until a constructor supplies it. Paper-level wrapper for the full clean-domain extension obligation of 'O_D^BS'.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Paper-level wrapper for the full clean-domain extension obligation of 'O_D^BS'. The paper clean domain contains every padded-zero source '|0>^(n-l)|s>^l|i>^n', including zero-amplitude sparse branches. QBE currently has only the active Lemma 1 image on valid row-dependent branches and a per-column interface for clean unused branches. This record lifts those pieces into one contract without choosing a reversible unused-branch image and without changing the active forward or dagger matrices.

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:2906. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.147●1 definition
  • structure(15 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.BandedSparseAccessFullCleanDomainExtensionContract :
  Type
    structure QuantumBlockEncoding.GHL2025.BandedSparseAccessFullCleanDomainExtensionContract :
  Type
    Paper-level wrapper for the full clean-domain extension obligation of
`O_D^BS`.

The paper clean domain contains every padded-zero source
`|0>^(n-l)|s>^l|i>^n`, including zero-amplitude sparse branches.  QBE currently
has only the active Lemma 1 image on valid row-dependent branches and a
per-column interface for clean unused branches.  This record lifts those
pieces into one contract without choosing a reversible unused-branch image and
without changing the active forward or dagger matrices.

    Fields

    sourceAnchor : String
    cleanInputPredicate : String
    validSparseBranchPredicate : String
    validCleanSourcePredicate : String
    unusedSparseBranchPredicate : String
    unusedBranchImageRuleContract : ℕ → QuantumBlockEncoding.GHL2025.BandedSparseAccessUnusedBranchImageRuleContract
    unusedBranchExtensionContract : ℕ → QuantumBlockEncoding.GHL2025.BandedSparseAccessUnusedBranchExtensionContract
    cleanDomainSplit : QuantumBlockEncoding.GHL2025.ObligationRecord
    validBranchAgreement : QuantumBlockEncoding.GHL2025.ObligationRecord
    unusedBranchImageSpecified : QuantumBlockEncoding.GHL2025.ObligationRecord
    unusedBranchImageFinite : QuantumBlockEncoding.GHL2025.ObligationRecord
    unusedBranchInjective : QuantumBlockEncoding.GHL2025.ObligationRecord
    fullCleanDomainInjective : QuantumBlockEncoding.GHL2025.ObligationRecord
    daggerCleanup : QuantumBlockEncoding.GHL2025.ObligationRecord
    unitaryExtension : QuantumBlockEncoding.GHL2025.ObligationRecord
Definition10.2.148
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access full clean domain extension contract”. Default full clean-domain extension contract for Lemma 1 'O_D^BS'.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Default full clean-domain extension contract for Lemma 1 'O_D^BS'. All semantic fields are false obligations. The nested per-column image-rule contract still has 'proposedImageIndex = none', so this declaration only records the missing proof interface for later source-domain reconciliation.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:2932. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.148●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessFullCleanDomainExtensionContract
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessFullCleanDomainExtensionContract
    Default full clean-domain extension contract for Lemma 1 `O_D^BS`.

All semantic fields are false obligations.  The nested per-column image-rule
contract still has `proposedImageIndex = none`, so this declaration only
records the missing proof interface for later source-domain reconciliation.
Theorem10.2.149
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access full clean domain extension contract flags false”; the hypotheses and conclusion in the code panel fix its exact scope. The full clean-domain wrapper is obligation-only in Phase 1.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The full clean-domain wrapper is obligation-only in Phase 1.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:2986. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.149●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract_flags_false
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
            p).cleanDomainSplit.proved =
      false ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
              p).validBranchAgreement.proved =
        false ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                p).unusedBranchImageSpecified.proved =
          false ∧
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                  p).unusedBranchImageFinite.proved =
            false ∧
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                    p).unusedBranchInjective.proved =
              false ∧
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                      p).fullCleanDomainInjective.proved =
                false ∧
              (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                        p).daggerCleanup.proved =
                  false ∧
                (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                          p).unitaryExtension.proved =
                    false ∧
                  ((QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                              p).unusedBranchImageRuleContract
                          j).proposedImageIndex =
                      none ∧
                    ((QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                                  p).unusedBranchImageRuleContract
                              j).imageSpecified.proved =
                        false ∧
                      ((QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                                  p).unusedBranchImageRuleContract
                              j).imageFinite.proved =
                        false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract_flags_false
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
            p).cleanDomainSplit.proved =
      false ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
              p).validBranchAgreement.proved =
        false ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                p).unusedBranchImageSpecified.proved =
          false ∧
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                  p).unusedBranchImageFinite.proved =
            false ∧
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                    p).unusedBranchInjective.proved =
              false ∧
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                      p).fullCleanDomainInjective.proved =
                false ∧
              (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                        p).daggerCleanup.proved =
                  false ∧
                (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                          p).unitaryExtension.proved =
                    false ∧
                  ((QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                              p).unusedBranchImageRuleContract
                          j).proposedImageIndex =
                      none ∧
                    ((QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                                  p).unusedBranchImageRuleContract
                              j).imageSpecified.proved =
                        false ∧
                      ((QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                                  p).unusedBranchImageRuleContract
                              j).imageFinite.proved =
                        false
    The full clean-domain wrapper is obligation-only in Phase 1.
Theorem10.2.150
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access full clean domain extension contract of unused branch”; the hypotheses and conclusion in the code panel fix its exact scope. The full clean-domain wrapper reuses the existing per-column unused-branch classifier bridge and keeps every extension proof flag false.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The full clean-domain wrapper reuses the existing per-column unused-branch classifier bridge and keeps every extension proof flag false.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3006. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.150●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract_of_unusedBranch
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ)
  (h :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperUnusedSparseBranch
        p j =
      true) :
  ((QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
              p).unusedBranchImageRuleContract
          j).cleanInput =
      true ∧
    ((QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                p).unusedBranchImageRuleContract
            j).validSparseBranch =
        false ∧
      ((QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                  p).unusedBranchImageRuleContract
              j).unusedSparseBranch =
          true ∧
        ((QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                    p).unusedBranchImageRuleContract
                j).proposedImageIndex =
            none ∧
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                    p).unusedBranchImageSpecified.proved =
              false ∧
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                      p).fullCleanDomainInjective.proved =
                false ∧
              (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                      p).unitaryExtension.proved =
                false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract_of_unusedBranch
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ)
  (h :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperUnusedSparseBranch
        p j =
      true) :
  ((QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
              p).unusedBranchImageRuleContract
          j).cleanInput =
      true ∧
    ((QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                p).unusedBranchImageRuleContract
            j).validSparseBranch =
        false ∧
      ((QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                  p).unusedBranchImageRuleContract
              j).unusedSparseBranch =
          true ∧
        ((QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                    p).unusedBranchImageRuleContract
                j).proposedImageIndex =
            none ∧
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                    p).unusedBranchImageSpecified.proved =
              false ∧
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                      p).fullCleanDomainInjective.proved =
                false ∧
              (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                      p).unitaryExtension.proved =
                false
    The full clean-domain wrapper reuses the existing per-column unused-branch
classifier bridge and keeps every extension proof flag false.
Theorem10.2.151
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access full clean domain extension contract local clean domain split”; the hypotheses and conclusion in the code panel fix its exact scope. Wrapper-facing form of the local clean-domain split audit.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Wrapper-facing form of the local clean-domain split audit. The classifier split is Lean-proved, while the full semantic wrapper still keeps 'cleanDomainSplit.proved = false' because no unused-branch image rule, injectivity proof, or unitary extension has been supplied.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3032. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.151●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract_localCleanDomainSplit
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput p j =
        true ↔
      QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource
            p j =
          true ∨
        QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperUnusedSparseBranch
            p j =
          true) ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
            p).cleanDomainSplit.proved =
      false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract_localCleanDomainSplit
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput
          p j =
        true ↔
      QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource
            p j =
          true ∨
        QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperUnusedSparseBranch
            p j =
          true) ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
            p).cleanDomainSplit.proved =
      false
    Wrapper-facing form of the local clean-domain split audit.

The classifier split is Lean-proved, while the full semantic wrapper still
keeps `cleanDomainSplit.proved = false` because no unused-branch image rule,
injectivity proof, or unitary extension has been supplied.
Definition10.2.152
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “banded sparse access unused zero branch source decision”. A proposition-valued field is a requirement until a constructor supplies it. Lean-facing source decision for unused zero-amplitude 'O_D^BS' branches.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Lean-facing source decision for unused zero-amplitude 'O_D^BS' branches. Cycle 14 records that no paper-backed image formula and no accepted external reversible-extension theorem currently supplies the missing image rule for clean unused sparse branches. This is a blocking dependency record, not a new oracle construction.

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:3049. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.152●1 definition
  • structure(6 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.BandedSparseAccessUnusedZeroBranchSourceDecision :
  Type
    structure QuantumBlockEncoding.GHL2025.BandedSparseAccessUnusedZeroBranchSourceDecision :
  Type
    Lean-facing source decision for unused zero-amplitude `O_D^BS` branches.

Cycle 14 records that no paper-backed image formula and no accepted external
reversible-extension theorem currently supplies the missing image rule for
clean unused sparse branches.  This is a blocking dependency record, not a new
oracle construction.

    Fields

    sourceAnchor : String
    citedResultKey : String
    paperImageRuleSpecified : Bool
    externalExtensionTheoremAccepted : Bool
    lowerProofSearchAllowed : Bool
    dependency : QuantumBlockEncoding.GHL2025.ObligationRecord
Definition10.2.153
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access unused zero branch source decision”. Default cycle-14 source decision for unused zero-amplitude sparse branches.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Default cycle-14 source decision for unused zero-amplitude sparse branches. The false Boolean fields deliberately prevent lower proof work from treating the current colliding active image as a permutation or unitary extension.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:3064. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.153●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessUnusedZeroBranchSourceDecision
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessUnusedZeroBranchSourceDecision
    Default cycle-14 source decision for unused zero-amplitude sparse branches.

The false Boolean fields deliberately prevent lower proof work from treating
the current colliding active image as a permutation or unitary extension.
Theorem10.2.154
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access unused zero branch source decision flags false”; the hypotheses and conclusion in the code panel fix its exact scope. The cycle-14 source decision is a blocking obligation, not a proof ticket.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The cycle-14 source decision is a blocking obligation, not a proof ticket.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3078. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.154●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision_flags_false :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision.paperImageRuleSpecified =
      false ∧
    QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision.externalExtensionTheoremAccepted =
        false ∧
      QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision.lowerProofSearchAllowed =
          false ∧
        QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision.dependency.proved =
          false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision_flags_false :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision.paperImageRuleSpecified =
      false ∧
    QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision.externalExtensionTheoremAccepted =
        false ∧
      QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision.lowerProofSearchAllowed =
          false ∧
        QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision.dependency.proved =
          false
    The cycle-14 source decision is a blocking obligation, not a proof ticket.
Theorem10.2.155
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access unused zero branch source decision keeps full domain flags false”; the hypotheses and conclusion in the code panel fix its exact scope. The source decision keeps the full clean-domain wrapper in obligation mode.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source decision keeps the full clean-domain wrapper in obligation mode. This ties the cited-results dependency to the existing wrapper fields without changing any active matrix or promoting any semantic proof flag.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3091. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.155●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision_keepsFullDomainFlagsFalse
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision.lowerProofSearchAllowed =
      false ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
              p).unusedBranchImageSpecified.proved =
        false ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                p).fullCleanDomainInjective.proved =
          false ∧
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                  p).daggerCleanup.proved =
            false ∧
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                  p).unitaryExtension.proved =
            false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision_keepsFullDomainFlagsFalse
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision.lowerProofSearchAllowed =
      false ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
              p).unusedBranchImageSpecified.proved =
        false ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                p).fullCleanDomainInjective.proved =
          false ∧
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                  p).daggerCleanup.proved =
            false ∧
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                  p).unitaryExtension.proved =
            false
    The source decision keeps the full clean-domain wrapper in obligation mode.

This ties the cited-results dependency to the existing wrapper fields without
changing any active matrix or promoting any semantic proof flag.
Theorem10.2.156
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access unused zero branch source decision keeps image rule unspecified”; the hypotheses and conclusion in the code panel fix its exact scope. The blocking source decision keeps every unused-branch image slot unspecified.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The blocking source decision keeps every unused-branch image slot unspecified. This is a guard for later lower packets: disabling proof search also means the per-column image-rule contract and the full-domain wrapper still expose 'proposedImageIndex = none' with false image-rule obligations.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3112. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.156●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision_keepsImageRuleUnspecified
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision.lowerProofSearchAllowed =
      false ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
            p j).proposedImageIndex =
        none ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
                p j).imageSpecified.proved =
          false ∧
        ((QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                    p).unusedBranchImageRuleContract
                j).proposedImageIndex =
            none ∧
          ((QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                        p).unusedBranchImageRuleContract
                    j).imageSpecified.proved =
              false ∧
            ((QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                        p).unusedBranchImageRuleContract
                    j).separatesActiveCollision.proved =
              false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision_keepsImageRuleUnspecified
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision.lowerProofSearchAllowed =
      false ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
            p j).proposedImageIndex =
        none ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
                p
                j).imageSpecified.proved =
          false ∧
        ((QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                    p).unusedBranchImageRuleContract
                j).proposedImageIndex =
            none ∧
          ((QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                        p).unusedBranchImageRuleContract
                    j).imageSpecified.proved =
              false ∧
            ((QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                        p).unusedBranchImageRuleContract
                    j).separatesActiveCollision.proved =
              false
    The blocking source decision keeps every unused-branch image slot unspecified.

This is a guard for later lower packets: disabling proof search also means the
per-column image-rule contract and the full-domain wrapper still expose
`proposedImageIndex = none` with false image-rule obligations.
Theorem10.2.157
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access unused zero branch source decision keeps paper contract flags false”; the hypotheses and conclusion in the code panel fix its exact scope. The blocking source decision also keeps the paper-level O_D^BS contract obligations false.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The blocking source decision also keeps the paper-level O_D^BS contract obligations false. This is separate from the full clean-domain wrapper: it pins the original Lemma 1 contract fields that a later proof packet would otherwise try to close.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3134. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.157●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision_keepsPaperContractFlagsFalse
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision.lowerProofSearchAllowed =
      false ∧
    (QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract
              p).forwardCorrect.proved =
        false ∧
      (QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract
                p).daggerCleanup.proved =
          false ∧
        (QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract
                p).unitaryExtension.proved =
          false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision_keepsPaperContractFlagsFalse
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision.lowerProofSearchAllowed =
      false ∧
    (QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract
              p).forwardCorrect.proved =
        false ∧
      (QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract
                p).daggerCleanup.proved =
          false ∧
        (QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract
                p).unitaryExtension.proved =
          false
    The blocking source decision also keeps the paper-level O_D^BS contract
obligations false.

This is separate from the full clean-domain wrapper: it pins the original
Lemma 1 contract fields that a later proof packet would otherwise try to close.
Definition10.2.158
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “banded sparse access robin zero inclusion source contract”. A proposition-valued field is a requirement until a constructor supplies it. Source transcript for the Robin zero-inclusion sentence near Theorem 1.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Source transcript for the Robin zero-inclusion sentence near Theorem 1. The paper states that zeros may be included in the sparse enumeration and then uses the range 's = 0, ..., kappa - 1' in Eq. ROBIN clarified. This source fact explains why the row-dependent nonzero-stencil classifier is not a faithful full domain restriction. It still does not choose an injective image for clean unused zero-amplitude branches.

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:3152. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.158●1 definition
  • structure(13 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.BandedSparseAccessRobinZeroInclusionSourceContract :
  Type
    structure QuantumBlockEncoding.GHL2025.BandedSparseAccessRobinZeroInclusionSourceContract :
  Type
    Source transcript for the Robin zero-inclusion sentence near Theorem 1.

The paper states that zeros may be included in the sparse enumeration and then
uses the range `s = 0, ..., kappa - 1` in Eq. ROBIN clarified.  This source
fact explains why the row-dependent nonzero-stencil classifier is not a
faithful full domain restriction.  It still does not choose an injective image
for clean unused zero-amplitude branches.

    Fields

    sourceAnchor : String
    zeroInclusionAnchor : String
    equationAnchor : String
    sparseIndexRange : String
    zerosIncludedInSparseEnumeration : Bool
    paperImageEquation : String
    unusedBranchImageRule : Option String
    unusedBranchImageIndex : Option ℕ
    reversibleExtensionTheorem : Option String
    closesUnusedZeroBranchExtension : Bool
    lowerProofSearchAllowed : Bool
    imageRuleObligation : QuantumBlockEncoding.GHL2025.ObligationRecord
    reversibleExtensionObligation : QuantumBlockEncoding.GHL2025.ObligationRecord
Definition10.2.159
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access robin zero inclusion source contract”. Default transcript of the GHL2025 Robin zero-inclusion source text.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Default transcript of the GHL2025 Robin zero-inclusion source text. The source records that zero-amplitude sparse branches remain in the kappa-wide sparse register. The missing image rule and any reversible extension theorem are intentionally absent, so this declaration does not unblock O_D^BS proof search.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:3176. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.159●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessRobinZeroInclusionSourceContract
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessRobinZeroInclusionSourceContract
    Default transcript of the GHL2025 Robin zero-inclusion source text.

The source records that zero-amplitude sparse branches remain in the
kappa-wide sparse register.  The missing image rule and any reversible
extension theorem are intentionally absent, so this declaration does not
unblock O_D^BS proof search.
Theorem10.2.160
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access robin zero inclusion source contract blocks unused zero branch”; the hypotheses and conclusion in the code panel fix its exact scope. The Robin zero-inclusion source transcript keeps the unused-branch route blocked.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The Robin zero-inclusion source transcript keeps the unused-branch route blocked. This guard records the exact source-backed inclusion of zero-amplitude sparse branches while preserving the absence of a branch image rule or reversible extension theorem.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3211. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.160●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract_blocks_unusedZeroBranch :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract.zerosIncludedInSparseEnumeration =
      true ∧
    QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract.unusedBranchImageRule =
        none ∧
      QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract.unusedBranchImageIndex =
          none ∧
        QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract.reversibleExtensionTheorem =
            none ∧
          QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract.closesUnusedZeroBranchExtension =
              false ∧
            QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract.lowerProofSearchAllowed =
                false ∧
              QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract.imageRuleObligation.proved =
                  false ∧
                QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract.reversibleExtensionObligation.proved =
                    false ∧
                  QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision.lowerProofSearchAllowed =
                    false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract_blocks_unusedZeroBranch :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract.zerosIncludedInSparseEnumeration =
      true ∧
    QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract.unusedBranchImageRule =
        none ∧
      QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract.unusedBranchImageIndex =
          none ∧
        QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract.reversibleExtensionTheorem =
            none ∧
          QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract.closesUnusedZeroBranchExtension =
              false ∧
            QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract.lowerProofSearchAllowed =
                false ∧
              QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract.imageRuleObligation.proved =
                  false ∧
                QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract.reversibleExtensionObligation.proved =
                    false ∧
                  QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision.lowerProofSearchAllowed =
                    false
    The Robin zero-inclusion source transcript keeps the unused-branch route
blocked.

This guard records the exact source-backed inclusion of zero-amplitude sparse
branches while preserving the absence of a branch image rule or reversible
extension theorem.
Theorem10.2.161
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access robin zero inclusion source contract keeps image rule unspecified”; the hypotheses and conclusion in the code panel fix its exact scope. The zero-inclusion transcript does not fill the per-column image-rule slot.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The zero-inclusion transcript does not fill the per-column image-rule slot. For every one-term parameter choice and source column, the direct image-rule contract remains unspecified while lower proof search is disabled.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3239. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.161●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract_keepsImageRuleUnspecified
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract.zerosIncludedInSparseEnumeration =
      true ∧
    QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract.unusedBranchImageIndex =
        none ∧
      QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract.reversibleExtensionTheorem =
          none ∧
        QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract.lowerProofSearchAllowed =
            false ∧
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
                  p j).proposedImageIndex =
              none ∧
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
                    p j).imageSpecified.proved =
              false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract_keepsImageRuleUnspecified
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract.zerosIncludedInSparseEnumeration =
      true ∧
    QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract.unusedBranchImageIndex =
        none ∧
      QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract.reversibleExtensionTheorem =
          none ∧
        QuantumBlockEncoding.GHL2025.bandedSparseAccessRobinZeroInclusionSourceContract.lowerProofSearchAllowed =
            false ∧
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
                  p
                  j).proposedImageIndex =
              none ∧
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
                    p
                    j).imageSpecified.proved =
              false
    The zero-inclusion transcript does not fill the per-column image-rule slot.

For every one-term parameter choice and source column, the direct image-rule
contract remains unspecified while lower proof search is disabled.
Definition10.2.162
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “banded sparse access prior pde source contract”. A proposition-valued field is a requirement until a constructor supplies it. Source contract imported from the prior PDE block-encoding paper.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Source contract imported from the prior PDE block-encoding paper. The prior paper supplies the same padded-register equation and an appendix decomposition into a first-row index unitary and modular addition. This record is intentionally only a source transcript: it does not provide a Robin-specific image rule for clean unused zero-amplitude sparse branches.

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:3264. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.162●1 definition
  • structure(10 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.BandedSparseAccessPriorPDESourceContract :
  Type
    structure QuantumBlockEncoding.GHL2025.BandedSparseAccessPriorPDESourceContract :
  Type
    Source contract imported from the prior PDE block-encoding paper.

The prior paper supplies the same padded-register equation and an appendix
decomposition into a first-row index unitary and modular addition.  This record
is intentionally only a source transcript: it does not provide a Robin-specific
image rule for clean unused zero-amplitude sparse branches.

    Fields

    sourceAnchor : String
    definitionAnchor : String
    lemmaAnchor : String
    appendixAnchor : String
    oracleEquation : String
    circuitDecomposition : String
    resourceClaim : QuantumBlockEncoding.GHL2025.ObligationRecord
    robinUnusedBranchImageRule : Option String
    closesUnusedZeroBranchExtension : Bool
    lowerProofSearchAllowed : Bool
Definition10.2.163
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access prior pde source contract”. Default transcript of arXiv:2405.12855v3 Definition 6, Lemma 1, and the appendix construction for 'O_A^BS'.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Default transcript of arXiv:2405.12855v3 Definition 6, Lemma 1, and the appendix construction for 'O_A^BS'. The field 'robinUnusedBranchImageRule = none' records the audit result: this source supports the imported sparse-access primitive, but not the missing Robin unused-branch image formula.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:3285. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.163●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPriorPDESourceContract :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessPriorPDESourceContract
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPriorPDESourceContract :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessPriorPDESourceContract
    Default transcript of arXiv:2405.12855v3 Definition 6, Lemma 1, and the
appendix construction for `O_A^BS`.

The field `robinUnusedBranchImageRule = none` records the audit result: this
source supports the imported sparse-access primitive, but not the missing
Robin unused-branch image formula.
Definition10.2.164
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “robin banded sparse access citation chain”. The explicit citation chain for the displayed Robin sparse-address equation.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The explicit citation chain for the displayed Robin sparse-address equation. Lemma 2 of arXiv:2506.20478 invokes Lemma 1 of arXiv:2405.12855; older transcript fields above retain their historical labels for drift auditing.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:3308. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.164●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.robinBandedSparseAccessCitationChain :
  List String
    def QuantumBlockEncoding.GHL2025.robinBandedSparseAccessCitationChain :
  List String
    The explicit citation chain for the displayed Robin sparse-address equation.
Lemma 2 of arXiv:2506.20478 invokes Lemma 1 of arXiv:2405.12855; older
transcript fields above retain their historical labels for drift auditing.
Theorem10.2.165
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “robin banded sparse access citation chain eq”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3313. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.165●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.robinBandedSparseAccessCitationChain_eq :
  QuantumBlockEncoding.GHL2025.robinBandedSparseAccessCitationChain =
    ["Guseynov-Huang-Liu 2025, arXiv:2506.20478, Lemma 2",
      "Guseynov-Huang-Liu 2024, arXiv:2405.12855v3, Lemma 1"]
    theorem QuantumBlockEncoding.GHL2025.robinBandedSparseAccessCitationChain_eq :
  QuantumBlockEncoding.GHL2025.robinBandedSparseAccessCitationChain =
    ["Guseynov-Huang-Liu 2025, arXiv:2506.20478, Lemma 2",
      "Guseynov-Huang-Liu 2024, arXiv:2405.12855v3, Lemma 1"]
Theorem10.2.166
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access prior pde source contract blocks unused zero branch”; the hypotheses and conclusion in the code panel fix its exact scope. The prior PDE source does not unblock the QBE unused-zero-branch extension.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The prior PDE source does not unblock the QBE unused-zero-branch extension. This is the compiled guard for the source audit: the cited theorem is recorded, but lower proof search for Robin unused-branch injectivity, cleanup, and unitarity remains disabled.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3326. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.166●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPriorPDESourceContract_blocks_unusedZeroBranch :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPriorPDESourceContract.robinUnusedBranchImageRule =
      none ∧
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPriorPDESourceContract.closesUnusedZeroBranchExtension =
        false ∧
      QuantumBlockEncoding.GHL2025.bandedSparseAccessPriorPDESourceContract.lowerProofSearchAllowed =
          false ∧
        QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision.lowerProofSearchAllowed =
          false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPriorPDESourceContract_blocks_unusedZeroBranch :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPriorPDESourceContract.robinUnusedBranchImageRule =
      none ∧
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPriorPDESourceContract.closesUnusedZeroBranchExtension =
        false ∧
      QuantumBlockEncoding.GHL2025.bandedSparseAccessPriorPDESourceContract.lowerProofSearchAllowed =
          false ∧
        QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision.lowerProofSearchAllowed =
          false
    The prior PDE source does not unblock the QBE unused-zero-branch extension.

This is the compiled guard for the source audit: the cited theorem is recorded,
but lower proof search for Robin unused-branch injectivity, cleanup, and
unitarity remains disabled.
Theorem10.2.167
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access prior pde source contract oracle equation”; the hypotheses and conclusion in the code panel fix its exact scope. The prior PDE source contract records the exact sparse-access equation.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The prior PDE source contract records the exact sparse-access equation.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3335. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.167●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPriorPDESourceContract_oracleEquation :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPriorPDESourceContract.oracleEquation =
    "O_A^BS |0>^(n-l)|s>^l|i>^n = |r_si>^n|i>^n"
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPriorPDESourceContract_oracleEquation :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPriorPDESourceContract.oracleEquation =
    "O_A^BS |0>^(n-l)|s>^l|i>^n = |r_si>^n|i>^n"
    The prior PDE source contract records the exact sparse-access equation.
Theorem10.2.168
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access prior pde source contract resource unproved”; the hypotheses and conclusion in the code panel fix its exact scope. The prior PDE resource claim remains an external obligation in QBE.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The prior PDE resource claim remains an external obligation in QBE.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3340. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.168●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPriorPDESourceContract_resource_unproved :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPriorPDESourceContract.resourceClaim.proved =
    false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPriorPDESourceContract_resource_unproved :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPriorPDESourceContract.resourceClaim.proved =
    false
    The prior PDE resource claim remains an external obligation in QBE.
Theorem10.2.169
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper clean input iff”; the hypotheses and conclusion in the code panel fix its exact scope. Boolean form of the Lemma 1 clean-input domain.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Boolean form of the Lemma 1 clean-input domain. The executable predicate is exactly the statement that the padded part of the 'O_D^BS' sparse-address register is zero. This only classifies columns; it does not prove the clean-input source equation or a unitary extension.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3350. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.169●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput_iff
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput p j =
      true ↔
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
          j).paddedZeroValue =
      0
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput_iff
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput
        p j =
      true ↔
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
          p j).paddedZeroValue =
      0
    Boolean form of the Lemma 1 clean-input domain.

The executable predicate is exactly the statement that the padded part of the
`O_D^BS` sparse-address register is zero.  This only classifies columns; it
does not prove the clean-input source equation or a unitary extension.
Definition10.2.170
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “banded sparse access paper column contract”. A proposition-valued field is a requirement until a constructor supplies it. Per-column audit record for the executable Lemma 1 paper image.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Per-column audit record for the executable Lemma 1 paper image. This records the source-domain flag, the image index, and the two register properties expected from the paper equation. The Boolean fields are executable checks for the current skeleton; they are not promoted to theorem-level correctness. The obligation fields keep the clean-domain and full-unitary extension gaps explicit.

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:3366. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.170●1 definition
  • structure(13 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.BandedSparseAccessPaperColumnContract :
  Type
    structure QuantumBlockEncoding.GHL2025.BandedSparseAccessPaperColumnContract :
  Type
    Per-column audit record for the executable Lemma 1 paper image.

This records the source-domain flag, the image index, and the two register
properties expected from the paper equation.  The Boolean fields are executable
checks for the current skeleton; they are not promoted to theorem-level
correctness.  The obligation fields keep the clean-domain and full-unitary
extension gaps explicit.

    Fields

    sourceAnchor : String
    inputRegisters : QuantumBlockEncoding.GHL2025.BandedSparseAccessPaperRegisters
    cleanInput : Bool
    imageIndex : ℕ
    imageRegisters : QuantumBlockEncoding.GHL2025.BandedSparseAccessPaperRegisters
    rowPreserved : Bool
    addressWritten : Bool
    addressInRange : Bool
    imageNoSpill : Bool
    cleanInputDomain : QuantumBlockEncoding.GHL2025.ObligationRecord
    addressRange : QuantumBlockEncoding.GHL2025.ObligationRecord
    noSpill : QuantumBlockEncoding.GHL2025.ObligationRecord
    unitaryExtension : QuantumBlockEncoding.GHL2025.ObligationRecord
Definition10.2.171
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access paper column contract”. Default per-column contract for the 'O_D^BS' paper image skeleton.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Default per-column contract for the 'O_D^BS' paper image skeleton.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:3385. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.171●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessPaperColumnContract
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessPaperColumnContract
    Default per-column contract for the `O_D^BS` paper image skeleton.
Theorem10.2.172
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper column contract input registers eq”; the hypotheses and conclusion in the code panel fix its exact scope. The per-column contract uses the shared Lemma 1 register extractor.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The per-column contract uses the shared Lemma 1 register extractor.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3409. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.172●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract_inputRegisters_eq
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract p
        j).inputRegisters =
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p j
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract_inputRegisters_eq
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract
        p j).inputRegisters =
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
      p j
    The per-column contract uses the shared Lemma 1 register extractor.
Theorem10.2.173
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper column contract clean input eq”; the hypotheses and conclusion in the code panel fix its exact scope. The per-column clean-domain flag is the executable padded-zero predicate.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The per-column clean-domain flag is the executable padded-zero predicate.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3415. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.173●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract_cleanInput_eq
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract p
        j).cleanInput =
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput p j
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract_cleanInput_eq
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract
        p j).cleanInput =
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput
      p j
    The per-column clean-domain flag is the executable padded-zero predicate.
Theorem10.2.174
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper column contract clean input iff”; the hypotheses and conclusion in the code panel fix its exact scope. The per-column clean-domain flag is true exactly on Lemma 1 clean columns.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The per-column clean-domain flag is true exactly on Lemma 1 clean columns. Columns with a nonzero padded register are still covered only by the explicit unitary-extension obligation.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3426. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.174●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract_cleanInput_iff
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract p
          j).cleanInput =
      true ↔
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
          j).paddedZeroValue =
      0
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract_cleanInput_iff
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract
          p j).cleanInput =
      true ↔
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
          p j).paddedZeroValue =
      0
    The per-column clean-domain flag is true exactly on Lemma 1 clean columns.

Columns with a nonzero padded register are still covered only by the explicit
unitary-extension obligation.
Theorem10.2.175
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper column contract unitary extension proved eq false”; the hypotheses and conclusion in the code panel fix its exact scope. The per-column audit keeps the full-space unitary extension as an open obligation for every column, including non-clean padded-register inputs.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The per-column audit keeps the full-space unitary extension as an open obligation for every column, including non-clean padded-register inputs.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3437. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.175●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract_unitaryExtension_proved_eq_false
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract p
          j).unitaryExtension.proved =
    false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract_unitaryExtension_proved_eq_false
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract
          p j).unitaryExtension.proved =
    false
    The per-column audit keeps the full-space unitary extension as an open
obligation for every column, including non-clean padded-register inputs.
Theorem10.2.176
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper column contract image index eq”; the hypotheses and conclusion in the code panel fix its exact scope. The per-column contract records the same image index as the paper-image skeleton.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The per-column contract records the same image index as the paper-image skeleton.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3442. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.176●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract_imageIndex_eq
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract p
        j).imageIndex =
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p j
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract_imageIndex_eq
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract
        p j).imageIndex =
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
      p j
    The per-column contract records the same image index as the paper-image skeleton.
Theorem10.2.177
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper column contract address in range eq”; the hypotheses and conclusion in the code panel fix its exact scope. The per-column contract records the executable n-bit address range check.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The per-column contract records the executable n-bit address range check.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3448. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.177●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract_addressInRange_eq
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract p
        j).addressInRange =
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddressInRange p
      j
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract_addressInRange_eq
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract
        p j).addressInRange =
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddressInRange
      p j
    The per-column contract records the executable n-bit address range check.
Theorem10.2.178
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper column contract image no spill eq”; the hypotheses and conclusion in the code panel fix its exact scope. The per-column contract records the executable high-bit no-spill check.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The per-column contract records the executable high-bit no-spill check.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3454. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.178●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract_imageNoSpill_eq
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract p
        j).imageNoSpill =
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageNoSpill p j
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract_imageNoSpill_eq
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract
        p j).imageNoSpill =
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageNoSpill
      p j
    The per-column contract records the executable high-bit no-spill check.
Theorem10.2.179
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper column contract row preserved eq true”; the hypotheses and conclusion in the code panel fix its exact scope. The per-column audit records that the paper image preserves the row register.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The per-column audit records that the paper image preserves the row register. This is an executable register-safety fact for the Phase 1 skeleton; it does not promote the paper-level 'forwardCorrect' obligation.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3465. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.179●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract_rowPreserved_eq_true
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract p
        j).rowPreserved =
    true
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract_rowPreserved_eq_true
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract
        p j).rowPreserved =
    true
    The per-column audit records that the paper image preserves the row register.

This is an executable register-safety fact for the Phase 1 skeleton; it does
not promote the paper-level `forwardCorrect` obligation.
Theorem10.2.180
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper column contract address written eq true of address lt”; the hypotheses and conclusion in the code panel fix its exact scope. The per-column audit records that the paper image writes the O_D register to the computed address whenever that address is an n-bit value.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The per-column audit records that the paper image writes the O_D register to the computed address whenever that address is an n-bit value.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3475. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.180●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract_addressWritten_eq_true_of_address_lt
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ)
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p j <
      1 <<< p.n) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract p
        j).addressWritten =
    true
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract_addressWritten_eq_true_of_address_lt
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ)
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
        p j <
      1 <<< p.n) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract
        p j).addressWritten =
    true
    The per-column audit records that the paper image writes the O_D register to
the computed address whenever that address is an n-bit value.
Theorem10.2.181
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper column contract address in range eq true of address lt”; the hypotheses and conclusion in the code panel fix its exact scope. The per-column address-range audit Boolean follows from the address bound.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The per-column address-range audit Boolean follows from the address bound.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3483. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.181●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract_addressInRange_eq_true_of_address_lt
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ)
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p j <
      1 <<< p.n) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract p
        j).addressInRange =
    true
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract_addressInRange_eq_true_of_address_lt
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ)
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
        p j <
      1 <<< p.n) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract
        p j).addressInRange =
    true
    The per-column address-range audit Boolean follows from the address bound.
Theorem10.2.182
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper column contract image no spill eq true of address lt”; the hypotheses and conclusion in the code panel fix its exact scope. The per-column no-spill audit Boolean follows from the address bound.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The per-column no-spill audit Boolean follows from the address bound.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3491. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.182●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract_imageNoSpill_eq_true_of_address_lt
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ)
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p j <
      1 <<< p.n) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract p
        j).imageNoSpill =
    true
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract_imageNoSpill_eq_true_of_address_lt
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ)
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
        p j <
      1 <<< p.n) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract
        p j).imageNoSpill =
    true
    The per-column no-spill audit Boolean follows from the address bound.
Theorem10.2.183
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper column contract register safety of address lt”; the hypotheses and conclusion in the code panel fix its exact scope. Reusable per-column register-safety package for the active Lemma 1 image skeleton.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Reusable per-column register-safety package for the active Lemma 1 image skeleton. The package is deliberately conditional on the existing n-bit address hypothesis, so it does not hide the paper parameter-family obligation.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3503. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.183●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract_registerSafety_of_address_lt
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ)
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p j <
      1 <<< p.n) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract p
          j).rowPreserved =
      true ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract
            p j).addressWritten =
        true ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract
              p j).addressInRange =
          true ∧
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract
              p j).imageNoSpill =
          true
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract_registerSafety_of_address_lt
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ)
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
        p j <
      1 <<< p.n) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract
          p j).rowPreserved =
      true ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract
            p j).addressWritten =
        true ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract
              p j).addressInRange =
          true ∧
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract
              p j).imageNoSpill =
          true
    Reusable per-column register-safety package for the active Lemma 1 image
skeleton.  The package is deliberately conditional on the existing n-bit
address hypothesis, so it does not hide the paper parameter-family obligation.
Definition10.2.184
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access paper matrix”. Matrix entries for the faithful Lemma 1 'O_D^BS' paper-image skeleton.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Matrix entries for the faithful Lemma 1 'O_D^BS' paper-image skeleton. The column 'j' has a candidate '1' entry at 'bandedSparseAccessPaperImage p j.val', which replaces the padded sparse-address register by 'r_si' and preserves the row register. This declaration is the active 'oneTermRobinGate_O_D_BS' matrix, but it does not prove that the image is in range, injective, unitary, or cleaned up by the dagger.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:3525. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.184●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperMatrix
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.Matrix
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
    QuantumBlockEncoding.Coeff
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperMatrix
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.Matrix
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p))
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p))
    QuantumBlockEncoding.Coeff
    Matrix entries for the faithful Lemma 1 `O_D^BS` paper-image skeleton.

The column `j` has a candidate `1` entry at
`bandedSparseAccessPaperImage p j.val`, which replaces the padded sparse-address
register by `r_si` and preserves the row register.  This declaration is the
active `oneTermRobinGate_O_D_BS` matrix, but it does not prove that the image is
in range, injective, unitary, or cleaned up by the dagger.
Theorem10.2.185
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper matrix eq image”; the hypotheses and conclusion in the code panel fix its exact scope. The paper-image matrix entry is governed by 'bandedSparseAccessPaperImage'.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The paper-image matrix entry is governed by 'bandedSparseAccessPaperImage'.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3531. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.185●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperMatrix_eq_image
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (i j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperMatrix p i j =
    if
        ↑i =
          QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p
            ↑j then
      QuantumBlockEncoding.Coeff.rat 1
    else QuantumBlockEncoding.Coeff.rat 0
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperMatrix_eq_image
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (i j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p))) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperMatrix
      p i j =
    if
        ↑i =
          QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
            p ↑j then
      QuantumBlockEncoding.Coeff.rat 1
    else QuantumBlockEncoding.Coeff.rat 0
    The paper-image matrix entry is governed by `bandedSparseAccessPaperImage`.
Theorem10.2.186
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper matrix image fin eq one”; the hypotheses and conclusion in the code panel fix its exact scope. Forward paper-image matrix entry at the finite image column.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Forward paper-image matrix entry at the finite image column. The hypotheses are the same explicit range hypotheses used to construct 'bandedSparseAccessPaperImageFin'. This theorem does not assert that the image function is injective or that the matrix is unitary.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3544. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.186●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperMatrix_imageFin_eq_one
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p ↑j <
      1 <<< p.n) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperMatrix p
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageFin p j
        haddr)
      j =
    QuantumBlockEncoding.Coeff.rat 1
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperMatrix_imageFin_eq_one
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
        p ↑j <
      1 <<< p.n) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperMatrix
      p
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageFin
        p j haddr)
      j =
    QuantumBlockEncoding.Coeff.rat 1
    Forward paper-image matrix entry at the finite image column.

The hypotheses are the same explicit range hypotheses used to construct
`bandedSparseAccessPaperImageFin`.  This theorem does not assert that the
image function is injective or that the matrix is unitary.
Definition10.2.187
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access paper dagger matrix”. Transpose-style matrix for the faithful Lemma 1 'O_D^BS' paper-image skeleton.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Transpose-style matrix for the faithful Lemma 1 'O_D^BS' paper-image skeleton. The entry 'M†[i,j]' is '1' exactly when column index 'j' is the forward paper image of row index 'i'. This is only the matrix-level transpose of the current executable image skeleton; the inverse, unitarity, and post-SWAP cleanup claims remain tracked by 'defaultBandedSparseAccessPaperContract p' with 'proved := false'.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:3561. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.187●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperDaggerMatrix
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.Matrix
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
    QuantumBlockEncoding.Coeff
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperDaggerMatrix
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.Matrix
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p))
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p))
    QuantumBlockEncoding.Coeff
    Transpose-style matrix for the faithful Lemma 1 `O_D^BS` paper-image skeleton.

The entry `M†[i,j]` is `1` exactly when column index `j` is the forward
paper image of row index `i`.  This is only the matrix-level transpose of the
current executable image skeleton; the inverse, unitarity, and post-SWAP cleanup
claims remain tracked by `defaultBandedSparseAccessPaperContract p` with
`proved := false`.
Theorem10.2.188
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper dagger matrix eq image”; the hypotheses and conclusion in the code panel fix its exact scope. The paper-image dagger matrix is the transpose-style matrix for the image skeleton.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The paper-image dagger matrix is the transpose-style matrix for the image skeleton.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3567. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.188●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperDaggerMatrix_eq_image
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (i j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperDaggerMatrix p i
      j =
    if
        ↑j =
          QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p
            ↑i then
      QuantumBlockEncoding.Coeff.rat 1
    else QuantumBlockEncoding.Coeff.rat 0
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperDaggerMatrix_eq_image
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (i j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p))) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperDaggerMatrix
      p i j =
    if
        ↑j =
          QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
            p ↑i then
      QuantumBlockEncoding.Coeff.rat 1
    else QuantumBlockEncoding.Coeff.rat 0
    The paper-image dagger matrix is the transpose-style matrix for the image skeleton.
Theorem10.2.189
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper dagger matrix image fin eq one”; the hypotheses and conclusion in the code panel fix its exact scope. Transpose-style paper-image matrix entry paired with the finite forward image.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Transpose-style paper-image matrix entry paired with the finite forward image. This is the entry relation needed before an inverse-on-range proof. It does not prove that the transpose-style matrix cleans the ancillas after SWAP.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3579. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.189●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperDaggerMatrix_imageFin_eq_one
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p ↑j <
      1 <<< p.n) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperDaggerMatrix p j
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageFin p j
        haddr) =
    QuantumBlockEncoding.Coeff.rat 1
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperDaggerMatrix_imageFin_eq_one
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
        p ↑j <
      1 <<< p.n) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperDaggerMatrix
      p j
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageFin
        p j haddr) =
    QuantumBlockEncoding.Coeff.rat 1
    Transpose-style paper-image matrix entry paired with the finite forward image.

This is the entry relation needed before an inverse-on-range proof.  It does
not prove that the transpose-style matrix cleans the ancillas after SWAP.
Definition10.2.190
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “robin sparse amplitude value”. Sparse amplitude value: the s-th nonzero stencil coefficient of row i in the Robin derivative matrix, returned as a Coeff value.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Sparse amplitude value: the s-th nonzero stencil coefficient of row i in the Robin derivative matrix, returned as a Coeff value. This is the data layer that both O_DT^S (sparse amplitude oracle, Lemma 3) and Ry_boundary (boundary-controlled rotations) need. The column index corresponding to each (s, i) pair is given by 'robinSparseColumnMap'. For the fourth-order central second-derivative stencil: - Bulk rows (K1 ≤ i ≤ K2): 5 entries at offsets {-2,-1,0,1,2} - Left boundary row 0: 3 entries with Robin correction (A1*dx term) - Left boundary row 1: 4 entries with Robin correction (A1*dx term) - Right boundary row N-2: 4 entries with Robin correction (B1*dx term) - Right boundary row N-1: 3 entries with Robin correction (B1*dx term) - Unused sparse indices (s ≥ entry count): Coeff.rat 0 main.tex:822-849, 1081-1083, 1113-1117 -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:3604. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.190●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.robinSparseAmplitudeValue (n s i : ℕ) :
  QuantumBlockEncoding.Coeff
    def QuantumBlockEncoding.GHL2025.robinSparseAmplitudeValue
  (n s i : ℕ) : QuantumBlockEncoding.Coeff
    Sparse amplitude value: the s-th nonzero stencil coefficient of row i in the
Robin derivative matrix, returned as a Coeff value.

This is the data layer that both O_DT^S (sparse amplitude oracle, Lemma 3)
and Ry_boundary (boundary-controlled rotations) need.  The column index
corresponding to each (s, i) pair is given by `robinSparseColumnMap`.

For the fourth-order central second-derivative stencil:
- Bulk rows (K1 ≤ i ≤ K2): 5 entries at offsets {-2,-1,0,1,2}
- Left boundary row 0: 3 entries with Robin correction (A1*dx term)
- Left boundary row 1: 4 entries with Robin correction (A1*dx term)
- Right boundary row N-2: 4 entries with Robin correction (B1*dx term)
- Right boundary row N-1: 3 entries with Robin correction (B1*dx term)
- Unused sparse indices (s ≥ entry count): Coeff.rat 0

main.tex:822-849, 1081-1083, 1113-1117 -
Definition10.2.191
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “robin global sparse amplitude value”. Global sparse-slot coefficient source for the one-term Robin table.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Global sparse-slot coefficient source for the one-term Robin table. Unlike 'robinSparseAmplitudeValue', the sparse index is interpreted through the active global slot table used by 'oneTermRobinGlobalSparseAddress'. Thus slot '2' is the zero-offset diagonal slot in every row. Boundary slots that are present in the global sparse register but absent from the Robin row carry coefficient '0'; the slot itself is not deleted. Guseynov-Huang-Liu 2025, Lemma 'Diagonal sparsity', Lemma 'Banded-sparse-access-oracle', the zero-inclusion paragraph before Theorem '1 term robin', and Eq. 'ROBIN clarified', arXiv:2506.20478.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:3661. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.191●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.robinGlobalSparseAmplitudeValue
  (n s i : ℕ) : QuantumBlockEncoding.Coeff
    def QuantumBlockEncoding.GHL2025.robinGlobalSparseAmplitudeValue
  (n s i : ℕ) : QuantumBlockEncoding.Coeff
    Global sparse-slot coefficient source for the one-term Robin table.

Unlike `robinSparseAmplitudeValue`, the sparse index is interpreted through
the active global slot table used by `oneTermRobinGlobalSparseAddress`.
Thus slot `2` is the zero-offset diagonal slot in every row. Boundary slots
that are present in the global sparse register but absent from the Robin row
carry coefficient `0`; the slot itself is not deleted.
Guseynov-Huang-Liu 2025, Lemma `Diagonal sparsity`, Lemma
`Banded-sparse-access-oracle`, the zero-inclusion paragraph before Theorem
`1 term robin`, and Eq. `ROBIN clarified`, arXiv:2506.20478.
Theorem10.2.192
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “robin global sparse amplitude value boundary slot 2 row 0 n 3”; the hypotheses and conclusion in the code panel fix its exact scope. Focused boundary regression: global slot '2' is the row-'0' diagonal.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Focused boundary regression: global slot '2' is the row-'0' diagonal.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3707. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.192●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.robinGlobalSparseAmplitudeValue_boundarySlot2_row0_n3 :
  QuantumBlockEncoding.GHL2025.robinGlobalSparseAmplitudeValue 3 2 0 =
    (QuantumBlockEncoding.Coeff.rat (-5 / 2)).add
      ((QuantumBlockEncoding.Coeff.rat (7 / 3)).mul
        (QuantumBlockEncoding.Coeff.symbol "A1*dx"))
    theorem QuantumBlockEncoding.GHL2025.robinGlobalSparseAmplitudeValue_boundarySlot2_row0_n3 :
  QuantumBlockEncoding.GHL2025.robinGlobalSparseAmplitudeValue
      3 2 0 =
    (QuantumBlockEncoding.Coeff.rat
          (-5 / 2)).add
      ((QuantumBlockEncoding.Coeff.rat
            (7 / 3)).mul
        (QuantumBlockEncoding.Coeff.symbol
          "A1*dx"))
    Focused boundary regression: global slot `2` is the row-`0` diagonal.
Theorem10.2.193
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “robin global sparse amplitude value boundary slot 2 differs row local n 3”; the hypotheses and conclusion in the code panel fix its exact scope. The focused global slot is not the old row-local sparse entry.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The focused global slot is not the old row-local sparse entry. This records the contract drift found by the gamma3 boundary packet without promoting any analytic normalizer or block-encoding flag.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3719. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.193●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.robinGlobalSparseAmplitudeValue_boundarySlot2_differs_rowLocal_n3 :
  QuantumBlockEncoding.GHL2025.robinGlobalSparseAmplitudeValue 3 2 0 ≠
    QuantumBlockEncoding.GHL2025.robinSparseAmplitudeValue 3 2 0
    theorem QuantumBlockEncoding.GHL2025.robinGlobalSparseAmplitudeValue_boundarySlot2_differs_rowLocal_n3 :
  QuantumBlockEncoding.GHL2025.robinGlobalSparseAmplitudeValue
      3 2 0 ≠
    QuantumBlockEncoding.GHL2025.robinSparseAmplitudeValue
      3 2 0
    The focused global slot is not the old row-local sparse entry.

This records the contract drift found by the gamma3 boundary packet without
promoting any analytic normalizer or block-encoding flag.
Definition10.2.194
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “derivative normalizer nd contract”. A proposition-valued field is a requirement until a constructor supplies it. Shared Phase-1 contract for every paper route that uses the normalized derivative coefficient 'D_j^(s) / N_D'.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Shared Phase-1 contract for every paper route that uses the normalized derivative coefficient 'D_j^(s) / N_D'. Both Lemma 3 'O_DT^S' and the boundary 'R_y' angle formulas use the same global sparse-slot coefficient source and the same normalizer symbol 'N_D'. This record keeps the common analytic gaps in one Lean object: nonzero normalizer, division semantics, coefficient bound, absolute-square semantics, square-root complement, arccos semantics, and two-by-two unitarity. It is a contract only; every obligation is false in Phase 1. Guseynov-Huang-Liu 2025, Lemma 3, Eq. (20), Fig. 1-term Robin, and boundary rotation equations, arXiv:2506.20478.

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:3737. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.194●1 definition
  • structure(14 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.DerivativeNormalizerNDContract : Type
    structure QuantumBlockEncoding.GHL2025.DerivativeNormalizerNDContract :
  Type
    Shared Phase-1 contract for every paper route that uses the normalized
derivative coefficient `D_j^(s) / N_D`.

Both Lemma 3 `O_DT^S` and the boundary `R_y` angle formulas use the same
global sparse-slot coefficient source and the same normalizer symbol `N_D`.
This record keeps the common analytic gaps in one Lean object: nonzero normalizer, division
semantics, coefficient bound, absolute-square semantics, square-root
complement, arccos semantics, and two-by-two unitarity.  It is a contract
only; every obligation is false in Phase 1.
Guseynov-Huang-Liu 2025, Lemma 3, Eq. (20), Fig. 1-term Robin, and boundary
rotation equations, arXiv:2506.20478.

    Fields

    sourceAnchor : String
    rowValue : ℕ
    sparseIndexValue : ℕ
    coefficient : QuantumBlockEncoding.Coeff
    normalizerND : QuantumBlockEncoding.Coeff
    normalizedCoefficient : QuantumBlockEncoding.Coeff
    normalizedCoefficientFormula : String
    nonzeroNormalizer : QuantumBlockEncoding.GHL2025.ObligationRecord
    divisionSemantics : QuantumBlockEncoding.GHL2025.ObligationRecord
    coefficientBound : QuantumBlockEncoding.GHL2025.ObligationRecord
    absSquareSemantics : QuantumBlockEncoding.GHL2025.ObligationRecord
    sqrtComplementSemantics : QuantumBlockEncoding.GHL2025.ObligationRecord
    arccosSemantics : QuantumBlockEncoding.GHL2025.ObligationRecord
    twoByTwoUnitary : QuantumBlockEncoding.GHL2025.ObligationRecord
Definition10.2.195
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “derivative normalizer nd contract”. Default shared 'N_D' normalizer contract for one Robin coefficient.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Default shared 'N_D' normalizer contract for one Robin coefficient. The normalized coefficient is represented by multiplying the sparse derivative coefficient by the formal symbol 'N_D_inv'. This is not a proof that 'N_D' is nonzero or that a division operation has been interpreted.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:3761. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.195●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  QuantumBlockEncoding.GHL2025.DerivativeNormalizerNDContract
    def QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  QuantumBlockEncoding.GHL2025.DerivativeNormalizerNDContract
    Default shared `N_D` normalizer contract for one Robin coefficient.

The normalized coefficient is represented by multiplying the sparse derivative
coefficient by the formal symbol `N_D_inv`.  This is not a proof that `N_D` is
nonzero or that a division operation has been interpreted.
Theorem10.2.196
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “derivative normalizer nd contract coefficient”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3808. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.196●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract_coefficient
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract p row
        sparse).coefficient =
    QuantumBlockEncoding.GHL2025.robinGlobalSparseAmplitudeValue p.n
      sparse row
    theorem QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract_coefficient
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
        p row sparse).coefficient =
    QuantumBlockEncoding.GHL2025.robinGlobalSparseAmplitudeValue
      p.n sparse row
Theorem10.2.197
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “derivative normalizer nd contract normalizer nd”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3813. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.197●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract_normalizerND
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract p row
        sparse).normalizerND =
    QuantumBlockEncoding.Coeff.symbol "N_D"
    theorem QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract_normalizerND
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
        p row sparse).normalizerND =
    QuantumBlockEncoding.Coeff.symbol
      "N_D"
Theorem10.2.198
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “derivative normalizer nd contract normalized coefficient”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3818. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.198●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract_normalizedCoefficient
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract p row
        sparse).normalizedCoefficient =
    (QuantumBlockEncoding.GHL2025.robinGlobalSparseAmplitudeValue p.n
          sparse row).mul
      (QuantumBlockEncoding.Coeff.symbol "N_D_inv")
    theorem QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract_normalizedCoefficient
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
        p row
        sparse).normalizedCoefficient =
    (QuantumBlockEncoding.GHL2025.robinGlobalSparseAmplitudeValue
          p.n sparse row).mul
      (QuantumBlockEncoding.Coeff.symbol
        "N_D_inv")
Definition10.2.199
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “derivative normalizer nd source bound”. A proposition-valued field is a requirement until a constructor supplies it. Phase-1 source/bound view for the shared 'N_D' normalizer contract.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Phase-1 source/bound view for the shared 'N_D' normalizer contract. This does not prove the analytic inequality. It only packages the exact coefficient source and the paper normalizer symbol used by the future bound obligation, so 'O_DT^S' and 'Ry_boundary' can point to the same fixed interface.

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:3831. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.199●1 definition
  • structure(7 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.DerivativeNormalizerNDSourceBound : Type
    structure QuantumBlockEncoding.GHL2025.DerivativeNormalizerNDSourceBound :
  Type
    Phase-1 source/bound view for the shared `N_D` normalizer contract.

This does not prove the analytic inequality.  It only packages the exact
coefficient source and the paper normalizer symbol used by the future bound
obligation, so `O_DT^S` and `Ry_boundary` can point to the same fixed
interface.

    Fields

    sourceAnchor : String
    rowValue : ℕ
    sparseIndexValue : ℕ
    sourceCoefficient : QuantumBlockEncoding.Coeff
    normalizerND : QuantumBlockEncoding.Coeff
    boundFormula : String
    coefficientBound : QuantumBlockEncoding.GHL2025.ObligationRecord
Definition10.2.200
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “derivative normalizer nd source bound”. Default source/bound interface for the paper statement '|D_j^(s)| <= N_D'.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Default source/bound interface for the paper statement '|D_j^(s)| <= N_D'. The coefficient and obligation are reused from 'derivativeNormalizerNDContract'; the obligation remains false until the coefficient semantics and analytic normalizer bound are formalized.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:3849. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.200●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  QuantumBlockEncoding.GHL2025.DerivativeNormalizerNDSourceBound
    def QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  QuantumBlockEncoding.GHL2025.DerivativeNormalizerNDSourceBound
    Default source/bound interface for the paper statement
`|D_j^(s)| <= N_D`.

The coefficient and obligation are reused from `derivativeNormalizerNDContract`;
the obligation remains false until the coefficient semantics and analytic
normalizer bound are formalized.
Theorem10.2.201
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “derivative normalizer nd source bound source coefficient”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3863. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.201●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound_sourceCoefficient
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound p row
        sparse).sourceCoefficient =
    QuantumBlockEncoding.GHL2025.robinGlobalSparseAmplitudeValue p.n
      sparse row
    theorem QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound_sourceCoefficient
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
        p row sparse).sourceCoefficient =
    QuantumBlockEncoding.GHL2025.robinGlobalSparseAmplitudeValue
      p.n sparse row
Theorem10.2.202
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “derivative normalizer nd source bound normalizer nd”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3868. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.202●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound_normalizerND
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound p row
        sparse).normalizerND =
    QuantumBlockEncoding.Coeff.symbol "N_D"
    theorem QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound_normalizerND
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
        p row sparse).normalizerND =
    QuantumBlockEncoding.Coeff.symbol
      "N_D"
Theorem10.2.203
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “derivative normalizer nd source bound bound formula”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3873. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.203●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound_boundFormula
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound p row
        sparse).boundFormula =
    "|D_j^(s)| <= N_D"
    theorem QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound_boundFormula
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
        p row sparse).boundFormula =
    "|D_j^(s)| <= N_D"
Theorem10.2.204
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “derivative normalizer nd source bound coefficient bound”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3878. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.204●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound_coefficientBound
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound p row
        sparse).coefficientBound =
    (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract p row
        sparse).coefficientBound
    theorem QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound_coefficientBound
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
        p row sparse).coefficientBound =
    (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
        p row sparse).coefficientBound
Theorem10.2.205
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “derivative normalizer nd source bound coefficient bound false”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3883. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.205●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound_coefficientBound_false
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound p row
          sparse).coefficientBound.proved =
    false
    theorem QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound_coefficientBound_false
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
          p row
          sparse).coefficientBound.proved =
    false
Definition10.2.206
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “indicator oracle matrix”. Honest U_indic matrix: controlled-X on the indicator qubit, conditioned on the system register being in the bulk window [K1, K2].

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Honest U_indic matrix: controlled-X on the indicator qubit, conditioned on the system register being in the bulk window [K1, K2]. For each basis state |j⟩: - Extract systemVal = bits [1, 1+n) of j - If K1 ≤ systemVal ≤ K2 (bulk row): flip indicator bit - Otherwise (boundary row): identity main.tex:1088-1099 -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:3896. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.206●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.indicatorOracleMatrix
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.Matrix
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
    QuantumBlockEncoding.Coeff
    def QuantumBlockEncoding.GHL2025.indicatorOracleMatrix
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.Matrix
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p))
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p))
    QuantumBlockEncoding.Coeff
    Honest U_indic matrix: controlled-X on the indicator qubit, conditioned on
the system register being in the bulk window [K1, K2].
For each basis state |j⟩:
  - Extract systemVal = bits [1, 1+n) of j
  - If K1 ≤ systemVal ≤ K2 (bulk row): flip indicator bit
  - Otherwise (boundary row): identity
main.tex:1088-1099 -
Definition10.2.207
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “one term robin gate u indic”. Gate matrix for U_indic using the honest permutation matrix.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Gate matrix for U_indic using the honest permutation matrix. Controlled-X on indicator bit at position 1+2n, conditioned on bulk membership. Unitarity proved: indicatorOracleMatrix_is_permutation shows each row and column has exactly one entry equal to 1, so the matrix is a permutation matrix (hence unitary). main.tex:1088-1099 -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:3914. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.207●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.oneTermRobinGate_U_indic
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GateMatrix QuantumBlockEncoding.Coeff
    (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)
    def QuantumBlockEncoding.GHL2025.oneTermRobinGate_U_indic
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GateMatrix
    QuantumBlockEncoding.Coeff
    (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
      p)
    Gate matrix for U_indic using the honest permutation matrix.
Controlled-X on indicator bit at position 1+2n, conditioned on bulk membership.
Unitarity proved: indicatorOracleMatrix_is_permutation shows each row and column
has exactly one entry equal to 1, so the matrix is a permutation matrix (hence unitary).
main.tex:1088-1099 -
Definition10.2.208
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “one term robin gate u indic dagger”. Theorem-facing Hermitian-conjugate slot for 'U_indic'.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Theorem-facing Hermitian-conjugate slot for 'U_indic'. The indicator permutation is self-inverse, so its dagger is represented by the same matrix. This gate record exists to keep the Fig. 1-term Robin transcript faithful; the active backend product is still the seven-gate list unless a separate theorem rewires it.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:3931. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.208●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.oneTermRobinGate_U_indic_dagger
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GateMatrix QuantumBlockEncoding.Coeff
    (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)
    def QuantumBlockEncoding.GHL2025.oneTermRobinGate_U_indic_dagger
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GateMatrix
    QuantumBlockEncoding.Coeff
    (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
      p)
    Theorem-facing Hermitian-conjugate slot for `U_indic`.

The indicator permutation is self-inverse, so its dagger is represented by the
same matrix.  This gate record exists to keep the Fig. 1-term Robin transcript
faithful; the active backend product is still the seven-gate list unless a
separate theorem rewires it.
Theorem10.2.209
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin gate u indic dagger matrix eq”; the hypotheses and conclusion in the code panel fix its exact scope. The theorem-facing 'U_indic^dagger' slot has the same matrix as 'U_indic'.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The theorem-facing 'U_indic^dagger' slot has the same matrix as 'U_indic'. This is only a transcript bridge. It does not insert the dagger slot into the active seven-gate backend product.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:3947. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.209●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGate_U_indic_dagger_matrix_eq
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  (QuantumBlockEncoding.GHL2025.oneTermRobinGate_U_indic_dagger
        p).matrix =
    (QuantumBlockEncoding.GHL2025.oneTermRobinGate_U_indic p).matrix
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGate_U_indic_dagger_matrix_eq
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  (QuantumBlockEncoding.GHL2025.oneTermRobinGate_U_indic_dagger
        p).matrix =
    (QuantumBlockEncoding.GHL2025.oneTermRobinGate_U_indic
        p).matrix
    The theorem-facing `U_indic^dagger` slot has the same matrix as `U_indic`.

This is only a transcript bridge.  It does not insert the dagger slot into the
active seven-gate backend product.
Definition10.2.210
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “sparse amplitude oracle dt matrix”. Honest O_DT^S diagonal matrix: encodes the sparse amplitude data on the diagonal for bulk rows (indicator=1) and acts as identity for boundary rows (indicator=0).

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Honest O_DT^S diagonal matrix: encodes the sparse amplitude data on the diagonal for bulk rows (indicator=1) and acts as identity for boundary rows (indicator=0). For each compound basis state |j⟩: - If indicator bit = 0 (boundary row): diagonal entry = Coeff.rat 1 (identity) - If indicator bit = 1 (bulk row): diagonal entry = robinSparseAmplitudeValue(n, s, i) - Off-diagonal entries are zero. NOTE: The paper's actual O_{D^T}^S (Lemma 3, main.tex:822-849) is a controlled rotation on the ancilla qubit, not a diagonal matrix. This diagonal encoding exercises the amplitude data pathway; the rotation structure is a proof obligation. main.tex:822-849 -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:3965. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.210●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTMatrix
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.Matrix
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
    QuantumBlockEncoding.Coeff
    def QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTMatrix
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.Matrix
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p))
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p))
    QuantumBlockEncoding.Coeff
    Honest O_DT^S diagonal matrix: encodes the sparse amplitude data on the diagonal
for bulk rows (indicator=1) and acts as identity for boundary rows (indicator=0).

For each compound basis state |j⟩:
  - If indicator bit = 0 (boundary row): diagonal entry = Coeff.rat 1 (identity)
  - If indicator bit = 1 (bulk row): diagonal entry = robinSparseAmplitudeValue(n, s, i)
  - Off-diagonal entries are zero.

NOTE: The paper's actual O_{D^T}^S (Lemma 3, main.tex:822-849) is a controlled
rotation on the ancilla qubit, not a diagonal matrix.  This diagonal encoding
exercises the amplitude data pathway; the rotation structure is a proof obligation.
main.tex:822-849 -
Definition10.2.211
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “sparse amplitude oracle dt paper registers”. A proposition-valued field is a requirement until a constructor supplies it. Register values used by the faithful Lemma 3 'O_DT^S' contract.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Register values used by the faithful Lemma 3 'O_DT^S' contract. The compound-index convention stores the rotation ancilla in bit 0, the system row in bits '[1, 1+n)', the padded sparse register in bits '[1+n, 1+2n)', and the indicator bit at 'robinIndicatorBitPosition p'. The 'nonAncillaValue' field is 'j >>> 1'; preserving it means that only the ancilla bit may change. Guseynov-Huang-Liu 2025, Lemma 3, arXiv:2506.20478.

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:3994. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.211●1 definition
  • structure(5 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.SparseAmplitudeOracleDTPaperRegisters :
  Type
    structure QuantumBlockEncoding.GHL2025.SparseAmplitudeOracleDTPaperRegisters :
  Type
    Register values used by the faithful Lemma 3 `O_DT^S` contract.

The compound-index convention stores the rotation ancilla in bit 0, the system
row in bits `[1, 1+n)`, the padded sparse register in bits `[1+n, 1+2n)`, and
the indicator bit at `robinIndicatorBitPosition p`.  The `nonAncillaValue`
field is `j >>> 1`; preserving it means that only the ancilla bit may change.
Guseynov-Huang-Liu 2025, Lemma 3, arXiv:2506.20478.

    Fields

    ancillaBit : ℕ
    indicatorBit : ℕ
    rowValue : ℕ
    sparseIndexValue : ℕ
    nonAncillaValue : ℕ
Definition10.2.212
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “sparse amplitude oracle dt paper registers”. Extract the Lemma 3 sparse-amplitude oracle registers from a compound basis index.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Extract the Lemma 3 sparse-amplitude oracle registers from a compound basis index. This is a source-contract skeleton for the paper's controlled rotation on the ancilla qubit; it leaves the legacy diagonal data helper available.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:4007. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.212●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTPaperRegisters
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  QuantumBlockEncoding.GHL2025.SparseAmplitudeOracleDTPaperRegisters
    def QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTPaperRegisters
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  QuantumBlockEncoding.GHL2025.SparseAmplitudeOracleDTPaperRegisters
    Extract the Lemma 3 sparse-amplitude oracle registers from a compound basis
index.  This is a source-contract skeleton for the paper's controlled rotation
on the ancilla qubit; it leaves the legacy diagonal data helper available.
Definition10.2.213
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “sparse amplitude oracle dt cos half”. Symbolic cosine half-angle entry for the Lemma 3 O_DT^S rotation.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Symbolic cosine half-angle entry for the Lemma 3 O_DT^S rotation.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:4025. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.213●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCosHalf
  (row sparse : ℕ) : QuantumBlockEncoding.Coeff
    def QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCosHalf
  (row sparse : ℕ) :
  QuantumBlockEncoding.Coeff
    Symbolic cosine half-angle entry for the Lemma 3 O_DT^S rotation.
Definition10.2.214
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “sparse amplitude oracle dt sin half”. Symbolic sine half-angle entry for the Lemma 3 O_DT^S rotation.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Symbolic sine half-angle entry for the Lemma 3 O_DT^S rotation.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:4029. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.214●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTSinHalf
  (row sparse : ℕ) : QuantumBlockEncoding.Coeff
    def QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTSinHalf
  (row sparse : ℕ) :
  QuantumBlockEncoding.Coeff
    Symbolic sine half-angle entry for the Lemma 3 O_DT^S rotation.
Definition10.2.215
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “sparse amplitude oracle dt coefficient normalizer obligation”. Explicit unresolved source obligation for the symbolic entries in the Lemma 3 'O_DT^S' rotation skeleton.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Explicit unresolved source obligation for the symbolic entries in the Lemma 3 'O_DT^S' rotation skeleton. Equation (20) of Guseynov-Huang-Liu 2025 maps '|0>|s>' to an amplitude whose '|0>' component is 'D^(s) / N_D' and whose complementary component is the square-root normalizer term. The Lean symbols 'sparseAmplitudeOracleDTCosHalf row sparse' and 'sparseAmplitudeOracleDTSinHalf row sparse' are only placeholders until this coefficient/normalizer relation and the corresponding two-by-two unitarity identity are formalized.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:4044. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.215●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerObligation :
  QuantumBlockEncoding.GHL2025.ObligationRecord
    def QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerObligation :
  QuantumBlockEncoding.GHL2025.ObligationRecord
    Explicit unresolved source obligation for the symbolic entries in the Lemma 3
`O_DT^S` rotation skeleton.

Equation (20) of Guseynov-Huang-Liu 2025 maps `|0>|s>` to an amplitude whose
`|0>` component is `D^(s) / N_D` and whose complementary component is the
square-root normalizer term.  The Lean symbols
`sparseAmplitudeOracleDTCosHalf row sparse` and
`sparseAmplitudeOracleDTSinHalf row sparse` are only placeholders until this
coefficient/normalizer relation and the corresponding two-by-two unitarity
identity are formalized.
Definition10.2.216
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “sparse amplitude oracle dt coefficient normalizer contract”. A proposition-valued field is a requirement until a constructor supplies it. Typed Eq.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Typed Eq. (20) coefficient-normalizer contract for one 'O_DT^S' rotation block. This binds the symbolic rotation entries to the concrete Robin sparse coefficient data and the paper's 'N_D' normalizer without proving the analytic identities. The three obligations stay false until Lean has a coefficient language with the required division, square-root, absolute-value, and unitarity facts.

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:4059. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.216●1 definition
  • structure(12 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.SparseAmplitudeOracleDTCoefficientNormalizerContract :
  Type
    structure QuantumBlockEncoding.GHL2025.SparseAmplitudeOracleDTCoefficientNormalizerContract :
  Type
    Typed Eq. (20) coefficient-normalizer contract for one `O_DT^S` rotation block.

This binds the symbolic rotation entries to the concrete Robin sparse
coefficient data and the paper's `N_D` normalizer without proving the analytic
identities.  The three obligations stay false until Lean has a coefficient
language with the required division, square-root, absolute-value, and unitarity
facts.

    Fields

    sourceAnchor : String
    rowValue : ℕ
    sparseIndexValue : ℕ
    coefficient : QuantumBlockEncoding.Coeff
    normalizerND : QuantumBlockEncoding.Coeff
    ketZeroEntry : QuantumBlockEncoding.Coeff
    ketOneEntry : QuantumBlockEncoding.Coeff
    ketZeroFormula : String
    ketOneFormula : String
    coefficientRelation : QuantumBlockEncoding.GHL2025.ObligationRecord
    complementRelation : QuantumBlockEncoding.GHL2025.ObligationRecord
    twoByTwoUnitary : QuantumBlockEncoding.GHL2025.ObligationRecord
Definition10.2.217
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “sparse amplitude oracle dt coefficient normalizer contract”. Default Eq.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Default Eq. (20) coefficient-normalizer contract for a Robin row and global sparse slot. The coefficient is 'robinGlobalSparseAmplitudeValue p.n sparse row'; the rotation entries are the symbols used by 'sparseAmplitudeOracleDTRotationMatrix'.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:4079. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.217●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerContract
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  QuantumBlockEncoding.GHL2025.SparseAmplitudeOracleDTCoefficientNormalizerContract
    def QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerContract
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  QuantumBlockEncoding.GHL2025.SparseAmplitudeOracleDTCoefficientNormalizerContract
    Default Eq. (20) coefficient-normalizer contract for a Robin row and global
sparse slot. The coefficient is `robinGlobalSparseAmplitudeValue p.n sparse row`; the
rotation entries are the symbols used by `sparseAmplitudeOracleDTRotationMatrix`.
Definition10.2.218
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “sparse amplitude oracle dt normalized coefficient”. Symbolic stand-in for the Lemma 3 normalized coefficient 'D_j^(s) / N_D'.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Symbolic stand-in for the Lemma 3 normalized coefficient 'D_j^(s) / N_D'. The factor 'Coeff.symbol "N_D_inv"' records the intended division by 'N_D'. It is not a proof that 'N_D' is nonzero or that the coefficient lies in the unit interval required by Eq. (20); those remain separate obligations.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:4114. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.218●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTNormalizedCoefficient
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) : QuantumBlockEncoding.Coeff
    def QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTNormalizedCoefficient
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  QuantumBlockEncoding.Coeff
    Symbolic stand-in for the Lemma 3 normalized coefficient `D_j^(s) / N_D`.

The factor `Coeff.symbol "N_D_inv"` records the intended division by `N_D`.
It is not a proof that `N_D` is nonzero or that the coefficient lies in the
unit interval required by Eq. (20); those remain separate obligations.
Definition10.2.219
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “sparse amplitude oracle dt coefficient normalizer proof route”. A proposition-valued field is a requirement until a constructor supplies it. Refined proof route for the 'odts_coeff_normalizer' block.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Refined proof route for the 'odts_coeff_normalizer' block. This record separates the typed Eq. (20) data from the analytic obligations: division by 'N_D', the paper's normalizer bound, the absolute-square term, the complementary square root, and the two-by-two unitarity identity. All proof obligations stay false in Phase 1.

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:4126. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.219●1 definition
  • structure(16 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.SparseAmplitudeOracleDTCoefficientNormalizerProofRoute :
  Type
    structure QuantumBlockEncoding.GHL2025.SparseAmplitudeOracleDTCoefficientNormalizerProofRoute :
  Type
    Refined proof route for the `odts_coeff_normalizer` block.

This record separates the typed Eq. (20) data from the analytic obligations:
division by `N_D`, the paper's normalizer bound, the absolute-square term, the
complementary square root, and the two-by-two unitarity identity.  All proof
obligations stay false in Phase 1.

    Fields

    sourceAnchor : String
    rowValue : ℕ
    sparseIndexValue : ℕ
    coefficient : QuantumBlockEncoding.Coeff
    normalizerND : QuantumBlockEncoding.Coeff
    normalizedCoefficient : QuantumBlockEncoding.Coeff
    normalizedCoefficientFormula : String
    ketZeroEntry : QuantumBlockEncoding.Coeff
    ketOneEntry : QuantumBlockEncoding.Coeff
    ketZeroFormula : String
    ketOneFormula : String
    coefficientDivision : QuantumBlockEncoding.GHL2025.ObligationRecord
    normalizerBound : QuantumBlockEncoding.GHL2025.ObligationRecord
    absSquareSemantics : QuantumBlockEncoding.GHL2025.ObligationRecord
    sqrtComplementSemantics : QuantumBlockEncoding.GHL2025.ObligationRecord
    twoByTwoUnitary : QuantumBlockEncoding.GHL2025.ObligationRecord
Definition10.2.220
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “sparse amplitude oracle dt coefficient normalizer proof route”. Default refined proof route for one 'O_DT^S' Eq.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Default refined proof route for one 'O_DT^S' Eq. (20) coefficient-normalizer block. The route keeps the construction fixed to the paper's controlled rotation and does not promote the gate-level unitarity claim.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:4150. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.220●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  QuantumBlockEncoding.GHL2025.SparseAmplitudeOracleDTCoefficientNormalizerProofRoute
    def QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  QuantumBlockEncoding.GHL2025.SparseAmplitudeOracleDTCoefficientNormalizerProofRoute
    Default refined proof route for one `O_DT^S` Eq. (20) coefficient-normalizer
block.  The route keeps the construction fixed to the paper's controlled
rotation and does not promote the gate-level unitarity claim.
Theorem10.2.221
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “sparse amplitude oracle dt coefficient normalizer proof route coefficient”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:4174. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.221●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute_coefficient
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
        p row sparse).coefficient =
    (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerContract
        p row sparse).coefficient
    theorem QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute_coefficient
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
        p row sparse).coefficient =
    (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerContract
        p row sparse).coefficient
Theorem10.2.222
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “sparse amplitude oracle dt coefficient normalizer proof route normalizer nd”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:4179. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.222●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute_normalizerND
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
        p row sparse).normalizerND =
    (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerContract
        p row sparse).normalizerND
    theorem QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute_normalizerND
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
        p row sparse).normalizerND =
    (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerContract
        p row sparse).normalizerND
Theorem10.2.223
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “sparse amplitude oracle dt coefficient normalizer proof route normalized coefficient”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:4184. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.223●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute_normalizedCoefficient
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
        p row sparse).normalizedCoefficient =
    QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTNormalizedCoefficient
      p row sparse
    theorem QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute_normalizedCoefficient
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
        p row
        sparse).normalizedCoefficient =
    QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTNormalizedCoefficient
      p row sparse
Theorem10.2.224
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “sparse amplitude oracle dt coefficient normalizer proof route shared nd”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:4189. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.224●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute_sharedND
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
          p row sparse).normalizerBound =
      (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract p row
          sparse).coefficientBound ∧
    (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
            p row sparse).coefficientDivision =
        (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract p
            row sparse).divisionSemantics ∧
      (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
              p row sparse).absSquareSemantics =
          (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract p
              row sparse).absSquareSemantics ∧
        (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
              p row sparse).sqrtComplementSemantics =
          (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract p
              row sparse).sqrtComplementSemantics
    theorem QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute_sharedND
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
          p row sparse).normalizerBound =
      (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
          p row sparse).coefficientBound ∧
    (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
            p row
            sparse).coefficientDivision =
        (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
            p row
            sparse).divisionSemantics ∧
      (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
              p row
              sparse).absSquareSemantics =
          (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
              p row
              sparse).absSquareSemantics ∧
        (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
              p row
              sparse).sqrtComplementSemantics =
          (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
              p row
              sparse).sqrtComplementSemantics
Theorem10.2.225
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “sparse amplitude oracle dt coefficient normalizer proof route source bound”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:4201. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.225●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute_sourceBound
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
          p row sparse).coefficient =
      (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound p
          row sparse).sourceCoefficient ∧
    (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
            p row sparse).normalizerND =
        (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
            p row sparse).normalizerND ∧
      (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
            p row sparse).normalizerBound =
        (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
            p row sparse).coefficientBound
    theorem QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute_sourceBound
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
          p row sparse).coefficient =
      (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
          p row
          sparse).sourceCoefficient ∧
    (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
            p row sparse).normalizerND =
        (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
            p row sparse).normalizerND ∧
      (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
            p row
            sparse).normalizerBound =
        (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
            p row sparse).coefficientBound
Theorem10.2.226
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “sparse amplitude oracle dt coefficient normalizer proof route ket zero entry”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:4211. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.226●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute_ketZeroEntry
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
        p row sparse).ketZeroEntry =
    (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerContract
        p row sparse).ketZeroEntry
    theorem QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute_ketZeroEntry
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
        p row sparse).ketZeroEntry =
    (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerContract
        p row sparse).ketZeroEntry
Theorem10.2.227
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “sparse amplitude oracle dt coefficient normalizer proof route ket one entry”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:4216. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.227●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute_ketOneEntry
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
        p row sparse).ketOneEntry =
    (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerContract
        p row sparse).ketOneEntry
    theorem QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute_ketOneEntry
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
        p row sparse).ketOneEntry =
    (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerContract
        p row sparse).ketOneEntry
Definition10.2.228
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “sparse amplitude oracle dt rotation matrix”. Faithful Lemma 3 controlled-rotation skeleton for 'O_DT^S'.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Faithful Lemma 3 controlled-rotation skeleton for 'O_DT^S'. For columns whose indicator bit is 0, the matrix acts as identity. For columns whose indicator bit is 1, it preserves every non-ancilla bit and applies a symbolic two-by-two rotation on ancilla bit 0. The symbols are indexed by the extracted row and sparse-index values; their connection to the Eq. (20) amplitudes determined by 'robinGlobalSparseAmplitudeValue p.n sparse row / N_D' remains the coefficient-normalizer proof obligation. Guseynov-Huang-Liu 2025, Lemma 3, arXiv:2506.20478.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:4231. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.228●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTRotationMatrix
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.Matrix
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
    QuantumBlockEncoding.Coeff
    def QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTRotationMatrix
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.Matrix
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p))
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p))
    QuantumBlockEncoding.Coeff
    Faithful Lemma 3 controlled-rotation skeleton for `O_DT^S`.

For columns whose indicator bit is 0, the matrix acts as identity.  For columns
whose indicator bit is 1, it preserves every non-ancilla bit and applies a
symbolic two-by-two rotation on ancilla bit 0.  The symbols are indexed by the
extracted row and sparse-index values; their connection to the Eq. (20)
amplitudes determined by `robinGlobalSparseAmplitudeValue p.n sparse row / N_D`
remains the coefficient-normalizer proof obligation.
Guseynov-Huang-Liu 2025, Lemma 3, arXiv:2506.20478.
Definition10.2.229
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “one term robin gate o dt s”. Gate matrix for O_DT^S using the faithful controlled-rotation skeleton.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Gate matrix for O_DT^S using the faithful controlled-rotation skeleton. The legacy diagonal helper 'sparseAmplitudeOracleDTMatrix' remains available as the coefficient-data path, but the active gate now preserves all non-ancilla bits and rotates bit 0 when the indicator bit is 1. Unitarity and the normalizer-bound trigonometric identity are not yet formally proved. main.tex:822-849 -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:4256. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.229●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_DT_S
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GateMatrix QuantumBlockEncoding.Coeff
    (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)
    def QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_DT_S
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GateMatrix
    QuantumBlockEncoding.Coeff
    (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
      p)
    Gate matrix for O_DT^S using the faithful controlled-rotation skeleton.
The legacy diagonal helper `sparseAmplitudeOracleDTMatrix` remains available as
the coefficient-data path, but the active gate now preserves all non-ancilla
bits and rotates bit 0 when the indicator bit is 1.  Unitarity and the
normalizer-bound trigonometric identity are not yet formally proved.
main.tex:822-849 -
Definition10.2.230
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “boundary rotation paper registers”. A proposition-valued field is a requirement until a constructor supplies it. Register values used by the faithful 'Ry_boundary' source contract.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Register values used by the faithful 'Ry_boundary' source contract. The compound-index convention is the same one used by the active matrix: ancilla bit 0 is the rotated qubit, bits '[1, 1+n)' contain the Robin row, the high part of the O_D register contains sparse index 's', and the indicator bit determines whether the boundary rotation is active. The 'nonAncillaValue' field is preserved by the controlled rotation. Guseynov-Huang-Liu 2025, Fig. 1-term Robin and Eq. angles for Ry, arXiv:2506.20478.

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:4275. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.230●1 definition
  • structure(5 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.BoundaryRotationPaperRegisters : Type
    structure QuantumBlockEncoding.GHL2025.BoundaryRotationPaperRegisters :
  Type
    Register values used by the faithful `Ry_boundary` source contract.

The compound-index convention is the same one used by the active matrix:
ancilla bit 0 is the rotated qubit, bits `[1, 1+n)` contain the Robin row,
the high part of the O_D register contains sparse index `s`, and the indicator
bit determines whether the boundary rotation is active.  The `nonAncillaValue`
field is preserved by the controlled rotation.
Guseynov-Huang-Liu 2025, Fig. 1-term Robin and Eq. angles for Ry,
arXiv:2506.20478.

    Fields

    ancillaBit : ℕ
    indicatorBit : ℕ
    rowValue : ℕ
    sparseIndexValue : ℕ
    nonAncillaValue : ℕ
Definition10.2.231
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “boundary rotation paper registers”. Extract the 'Ry_boundary' register fields from a compound basis index.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Extract the 'Ry_boundary' register fields from a compound basis index. This is a source-contract skeleton; it does not prove the angle identities or unitarity of the symbolic rotation block.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:4288. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.231●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.boundaryRotationPaperRegisters
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  QuantumBlockEncoding.GHL2025.BoundaryRotationPaperRegisters
    def QuantumBlockEncoding.GHL2025.boundaryRotationPaperRegisters
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  QuantumBlockEncoding.GHL2025.BoundaryRotationPaperRegisters
    Extract the `Ry_boundary` register fields from a compound basis index.
This is a source-contract skeleton; it does not prove the angle identities or
unitarity of the symbolic rotation block.
Definition10.2.232
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “boundary rotation cos half”. Symbolic cosine half-angle entry for the 'Ry_boundary' rotation.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Symbolic cosine half-angle entry for the 'Ry_boundary' rotation.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:4306. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.232●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.boundaryRotationCosHalf (row sparse : ℕ) :
  QuantumBlockEncoding.Coeff
    def QuantumBlockEncoding.GHL2025.boundaryRotationCosHalf
  (row sparse : ℕ) :
  QuantumBlockEncoding.Coeff
    Symbolic cosine half-angle entry for the `Ry_boundary` rotation.
Definition10.2.233
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “boundary rotation sin half”. Symbolic sine half-angle entry for the 'Ry_boundary' rotation.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Symbolic sine half-angle entry for the 'Ry_boundary' rotation.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:4310. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.233●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.boundaryRotationSinHalf (row sparse : ℕ) :
  QuantumBlockEncoding.Coeff
    def QuantumBlockEncoding.GHL2025.boundaryRotationSinHalf
  (row sparse : ℕ) :
  QuantumBlockEncoding.Coeff
    Symbolic sine half-angle entry for the `Ry_boundary` rotation.
Definition10.2.234
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “boundary rotation angle normalizer obligation”. Explicit unresolved source obligation for the 'Ry_boundary' angle/normalizer relation.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Explicit unresolved source obligation for the 'Ry_boundary' angle/normalizer relation. The paper uses angles 'theta_j^s = arccos(D_j^(s) / N_D)' for boundary rows. The Lean symbols 'boundaryRotationCosHalf row sparse' and 'boundaryRotationSinHalf row sparse' are placeholders until the half-angle identities and the two-by-two unitarity relation are formalized.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:4322. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.234●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerObligation :
  QuantumBlockEncoding.GHL2025.ObligationRecord
    def QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerObligation :
  QuantumBlockEncoding.GHL2025.ObligationRecord
    Explicit unresolved source obligation for the `Ry_boundary` angle/normalizer
relation.

The paper uses angles `theta_j^s = arccos(D_j^(s) / N_D)` for boundary rows.
The Lean symbols `boundaryRotationCosHalf row sparse` and
`boundaryRotationSinHalf row sparse` are placeholders until the half-angle
identities and the two-by-two unitarity relation are formalized.
Definition10.2.235
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “boundary rotation angle normalizer contract”. A proposition-valued field is a requirement until a constructor supplies it. Typed angle/normalizer contract for one 'Ry_boundary' rotation block.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Typed angle/normalizer contract for one 'Ry_boundary' rotation block. This binds the symbolic half-angle entries used by 'boundaryRotationMatrix' to the Robin sparse coefficient source and the paper normalizer 'N_D'. It records the exact obligations without asserting the arccos relation, half-angle formulas, control condition, or unitarity.

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:4336. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.235●1 definition
  • structure(15 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.BoundaryRotationAngleNormalizerContract :
  Type
    structure QuantumBlockEncoding.GHL2025.BoundaryRotationAngleNormalizerContract :
  Type
    Typed angle/normalizer contract for one `Ry_boundary` rotation block.

This binds the symbolic half-angle entries used by `boundaryRotationMatrix` to
the Robin sparse coefficient source and the paper normalizer `N_D`.  It records
the exact obligations without asserting the arccos relation, half-angle
formulas, control condition, or unitarity.

    Fields

    sourceAnchor : String
    rowValue : ℕ
    sparseIndexValue : ℕ
    coefficient : QuantumBlockEncoding.Coeff
    normalizerND : QuantumBlockEncoding.Coeff
    thetaFormula : String
    cosHalfEntry : QuantumBlockEncoding.Coeff
    sinHalfEntry : QuantumBlockEncoding.Coeff
    cosHalfFormula : String
    sinHalfFormula : String
    boundaryControl : QuantumBlockEncoding.GHL2025.ObligationRecord
    arccosArgumentRelation : QuantumBlockEncoding.GHL2025.ObligationRecord
    cosHalfRelation : QuantumBlockEncoding.GHL2025.ObligationRecord
    sinHalfRelation : QuantumBlockEncoding.GHL2025.ObligationRecord
    twoByTwoUnitary : QuantumBlockEncoding.GHL2025.ObligationRecord
Definition10.2.236
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “boundary rotation angle normalizer contract”. Default 'Ry_boundary' angle/normalizer contract for one Robin row and global sparse slot.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Default 'Ry_boundary' angle/normalizer contract for one Robin row and global sparse slot. The coefficient is 'robinGlobalSparseAmplitudeValue p.n sparse row'; the rotation entries are the symbols used by 'boundaryRotationMatrix'.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:4359. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.236●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerContract
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  QuantumBlockEncoding.GHL2025.BoundaryRotationAngleNormalizerContract
    def QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerContract
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  QuantumBlockEncoding.GHL2025.BoundaryRotationAngleNormalizerContract
    Default `Ry_boundary` angle/normalizer contract for one Robin row and global
sparse slot. The coefficient is `robinGlobalSparseAmplitudeValue p.n sparse row`; the
rotation entries are the symbols used by `boundaryRotationMatrix`.
Theorem10.2.237
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “boundary rotation angle normalizer contract coefficient”; the hypotheses and conclusion in the code panel fix its exact scope. The coefficient source of the 'Ry_boundary' angle contract is definitionally the Robin global sparse-slot amplitude data layer.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The coefficient source of the 'Ry_boundary' angle contract is definitionally the Robin global sparse-slot amplitude data layer.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:4402. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.237●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerContract_coefficient
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerContract
        p row sparse).coefficient =
    QuantumBlockEncoding.GHL2025.robinGlobalSparseAmplitudeValue p.n
      sparse row
    theorem QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerContract_coefficient
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerContract
        p row sparse).coefficient =
    QuantumBlockEncoding.GHL2025.robinGlobalSparseAmplitudeValue
      p.n sparse row
    The coefficient source of the `Ry_boundary` angle contract is definitionally
the Robin global sparse-slot amplitude data layer.
Definition10.2.238
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “boundary rotation normalized coefficient”. Symbolic stand-in for the paper argument 'D_j^(s) / N_D'.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Symbolic stand-in for the paper argument 'D_j^(s) / N_D'. The factor 'Coeff.symbol "N_D_inv"' is not a proof that 'N_D' is invertible. It only records the intended normalized coefficient while the required division semantics and nonzero normalizer condition remain explicit obligations.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:4414. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.238●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.boundaryRotationNormalizedCoefficient
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) : QuantumBlockEncoding.Coeff
    def QuantumBlockEncoding.GHL2025.boundaryRotationNormalizedCoefficient
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  QuantumBlockEncoding.Coeff
    Symbolic stand-in for the paper argument `D_j^(s) / N_D`.

The factor `Coeff.symbol "N_D_inv"` is not a proof that `N_D` is invertible.
It only records the intended normalized coefficient while the required division
semantics and nonzero normalizer condition remain explicit obligations.
Definition10.2.239
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “boundary rotation angle normalizer proof route”. A proposition-valued field is a requirement until a constructor supplies it. Refined proof route for the 'ryb_angle_normalizer' block.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Refined proof route for the 'ryb_angle_normalizer' block. This record separates the typed data already present in 'BoundaryRotationAngleNormalizerContract' from the missing analytic semantics: division by 'N_D', real arccos, square roots, the paper's normalizer bound, and the resulting two-by-two unitarity identity. All proof obligations stay false in Phase 1.

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:4427. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.239●1 definition
  • structure(17 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.BoundaryRotationAngleNormalizerProofRoute :
  Type
    structure QuantumBlockEncoding.GHL2025.BoundaryRotationAngleNormalizerProofRoute :
  Type
    Refined proof route for the `ryb_angle_normalizer` block.

This record separates the typed data already present in
`BoundaryRotationAngleNormalizerContract` from the missing analytic semantics:
division by `N_D`, real arccos, square roots, the paper's normalizer bound, and
the resulting two-by-two unitarity identity.  All proof obligations stay false
in Phase 1.

    Fields

    sourceAnchor : String
    rowValue : ℕ
    sparseIndexValue : ℕ
    coefficient : QuantumBlockEncoding.Coeff
    normalizerND : QuantumBlockEncoding.Coeff
    arccosArgument : QuantumBlockEncoding.Coeff
    arccosArgumentFormula : String
    thetaFormula : String
    cosHalfEntry : QuantumBlockEncoding.Coeff
    sinHalfEntry : QuantumBlockEncoding.Coeff
    cosHalfFormula : String
    sinHalfFormula : String
    coefficientDivision : QuantumBlockEncoding.GHL2025.ObligationRecord
    realArccosSemantics : QuantumBlockEncoding.GHL2025.ObligationRecord
    halfAngleSemantics : QuantumBlockEncoding.GHL2025.ObligationRecord
    normalizerBound : QuantumBlockEncoding.GHL2025.ObligationRecord
    twoByTwoUnitary : QuantumBlockEncoding.GHL2025.ObligationRecord
Definition10.2.240
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “boundary rotation angle normalizer proof route”. Default refined proof route for one 'Ry_boundary' angle-normalizer block.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Default refined proof route for one 'Ry_boundary' angle-normalizer block. The route keeps the construction fixed to the paper formula 'theta_j^s = arccos(D_j^(s) / N_D)'. It does not introduce a replacement angle or promote the gate-level unitarity claim.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:4454. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.240●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  QuantumBlockEncoding.GHL2025.BoundaryRotationAngleNormalizerProofRoute
    def QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  QuantumBlockEncoding.GHL2025.BoundaryRotationAngleNormalizerProofRoute
    Default refined proof route for one `Ry_boundary` angle-normalizer block.

The route keeps the construction fixed to the paper formula
`theta_j^s = arccos(D_j^(s) / N_D)`.  It does not introduce a replacement angle
or promote the gate-level unitarity claim.
Theorem10.2.241
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “boundary rotation angle normalizer proof route coefficient”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:4483. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.241●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute_coefficient
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
        p row sparse).coefficient =
    (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerContract
        p row sparse).coefficient
    theorem QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute_coefficient
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
        p row sparse).coefficient =
    (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerContract
        p row sparse).coefficient
Theorem10.2.242
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “boundary rotation angle normalizer proof route arccos argument”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:4488. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.242●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute_arccosArgument
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
        p row sparse).arccosArgument =
    QuantumBlockEncoding.GHL2025.boundaryRotationNormalizedCoefficient p
      row sparse
    theorem QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute_arccosArgument
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
        p row sparse).arccosArgument =
    QuantumBlockEncoding.GHL2025.boundaryRotationNormalizedCoefficient
      p row sparse
Theorem10.2.243
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “boundary rotation angle normalizer proof route shared nd”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:4493. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.243●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute_sharedND
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
          p row sparse).normalizerBound =
      (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract p row
          sparse).coefficientBound ∧
    (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
            p row sparse).coefficientDivision =
        (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract p
            row sparse).divisionSemantics ∧
      (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
            p row sparse).realArccosSemantics =
        (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract p
            row sparse).arccosSemantics
    theorem QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute_sharedND
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
          p row sparse).normalizerBound =
      (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
          p row sparse).coefficientBound ∧
    (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
            p row
            sparse).coefficientDivision =
        (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
            p row
            sparse).divisionSemantics ∧
      (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
            p row
            sparse).realArccosSemantics =
        (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
            p row sparse).arccosSemantics
Theorem10.2.244
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “boundary rotation angle normalizer proof route source bound”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:4503. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.244●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute_sourceBound
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
          p row sparse).coefficient =
      (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound p
          row sparse).sourceCoefficient ∧
    (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
            p row sparse).normalizerND =
        (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
            p row sparse).normalizerND ∧
      (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
            p row sparse).normalizerBound =
        (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
            p row sparse).coefficientBound
    theorem QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute_sourceBound
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
          p row sparse).coefficient =
      (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
          p row
          sparse).sourceCoefficient ∧
    (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
            p row sparse).normalizerND =
        (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
            p row sparse).normalizerND ∧
      (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
            p row
            sparse).normalizerBound =
        (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
            p row sparse).coefficientBound
Theorem10.2.245
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “derivative normalizer nd source bound shared routes”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:4513. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.245●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound_sharedRoutes
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
          p row sparse).coefficient =
      (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
          p row sparse).coefficient ∧
    (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
            p row sparse).normalizerND =
        (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
            p row sparse).normalizerND ∧
      (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
            p row sparse).normalizerBound =
        (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
            p row sparse).normalizerBound
    theorem QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound_sharedRoutes
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
          p row sparse).coefficient =
      (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
          p row sparse).coefficient ∧
    (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
            p row sparse).normalizerND =
        (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
            p row sparse).normalizerND ∧
      (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
            p row
            sparse).normalizerBound =
        (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
            p row sparse).normalizerBound
Theorem10.2.246
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “robin global sparse amplitude value shared normalizer routes”; the hypotheses and conclusion in the code panel fix its exact scope. Bridge showing that the shared 'N_D' route is now sourced from the active global sparse-slot coefficient table.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Bridge showing that the shared 'N_D' route is now sourced from the active global sparse-slot coefficient table. This only wires coefficient data through the existing contracts. It does not prove the analytic division, arccos, half-angle, or unitarity obligations.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:4530. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.246●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.robinGlobalSparseAmplitudeValue_sharedNormalizerRoutes
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound p row
          sparse).sourceCoefficient =
      QuantumBlockEncoding.GHL2025.robinGlobalSparseAmplitudeValue p.n
        sparse row ∧
    (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
            p row sparse).coefficient =
        QuantumBlockEncoding.GHL2025.robinGlobalSparseAmplitudeValue p.n
          sparse row ∧
      (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
              p row sparse).coefficient =
          QuantumBlockEncoding.GHL2025.robinGlobalSparseAmplitudeValue
            p.n sparse row ∧
        (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
              p row sparse).coefficient =
          (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
              p row sparse).coefficient
    theorem QuantumBlockEncoding.GHL2025.robinGlobalSparseAmplitudeValue_sharedNormalizerRoutes
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
          p row
          sparse).sourceCoefficient =
      QuantumBlockEncoding.GHL2025.robinGlobalSparseAmplitudeValue
        p.n sparse row ∧
    (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
            p row sparse).coefficient =
        QuantumBlockEncoding.GHL2025.robinGlobalSparseAmplitudeValue
          p.n sparse row ∧
      (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
              p row sparse).coefficient =
          QuantumBlockEncoding.GHL2025.robinGlobalSparseAmplitudeValue
            p.n sparse row ∧
        (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
              p row sparse).coefficient =
          (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
              p row sparse).coefficient
    Bridge showing that the shared `N_D` route is now sourced from the active
global sparse-slot coefficient table.

This only wires coefficient data through the existing contracts. It does not
prove the analytic division, arccos, half-angle, or unitarity obligations.
Definition10.2.247
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “boundary rotation matrix”. Honest Ry_boundary matrix: controlled R_y rotation on the ancilla qubit (bit 0), conditioned on the indicator bit being 0 (boundary row).

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Honest Ry_boundary matrix: controlled R_y rotation on the ancilla qubit (bit 0), conditioned on the indicator bit being 0 (boundary row). For bulk rows (indicator=1): acts as identity (no rotation). For boundary rows (indicator=0): applies R_y(θ_j^s) on the ancilla qubit, where θ_j^s = arccos(D_j^(s) / N_D) (main.tex:1115-1120, Eq. angles for Ry). The R_y(θ) matrix on the ancilla qubit: M(|0⟩, |0⟩) = cos(θ/2), M(|1⟩, |0⟩) = sin(θ/2) M(|0⟩, |1⟩) = -sin(θ/2), M(|1⟩, |1⟩) = cos(θ/2) Rotation entries are symbolic since the exact trigonometric values involve square roots: cos(θ/2) = √((1 + D/N_D)/2), sin(θ/2) = √((1 - D/N_D)/2). main.tex:1115-1120 -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:4557. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.247●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.boundaryRotationMatrix
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.Matrix
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
    QuantumBlockEncoding.Coeff
    def QuantumBlockEncoding.GHL2025.boundaryRotationMatrix
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.Matrix
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p))
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p))
    QuantumBlockEncoding.Coeff
    Honest Ry_boundary matrix: controlled R_y rotation on the ancilla qubit (bit 0),
conditioned on the indicator bit being 0 (boundary row).

For bulk rows (indicator=1): acts as identity (no rotation).
For boundary rows (indicator=0): applies R_y(θ_j^s) on the ancilla qubit,
where θ_j^s = arccos(D_j^(s) / N_D) (main.tex:1115-1120, Eq. angles for Ry).

The R_y(θ) matrix on the ancilla qubit:
  M(|0⟩, |0⟩) = cos(θ/2),  M(|1⟩, |0⟩) = sin(θ/2)
  M(|0⟩, |1⟩) = -sin(θ/2), M(|1⟩, |1⟩) = cos(θ/2)

Rotation entries are symbolic since the exact trigonometric values involve
square roots: cos(θ/2) = √((1 + D/N_D)/2), sin(θ/2) = √((1 - D/N_D)/2).
main.tex:1115-1120 -
Definition10.2.248
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “one term robin gate ry boundary”. Gate matrix for Ry_boundary using the honest controlled rotation matrix.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Gate matrix for Ry_boundary using the honest controlled rotation matrix. R_y rotation on the ancilla qubit for boundary rows (indicator=0); identity for bulk rows (indicator=1). Unitarity not yet formally proved. main.tex:1115-1120 -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:4592. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.248●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.oneTermRobinGate_Ry_boundary
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GateMatrix QuantumBlockEncoding.Coeff
    (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)
    def QuantumBlockEncoding.GHL2025.oneTermRobinGate_Ry_boundary
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GateMatrix
    QuantumBlockEncoding.Coeff
    (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
      p)
    Gate matrix for Ry_boundary using the honest controlled rotation matrix.
R_y rotation on the ancilla qubit for boundary rows (indicator=0);
identity for bulk rows (indicator=1).
Unitarity not yet formally proved.
main.tex:1115-1120 -
Theorem10.2.249
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “derivative normalizer nd shared route flags false”; the hypotheses and conclusion in the code panel fix its exact scope. Guard for the shared 'N_D' Phase-1 route.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Guard for the shared 'N_D' Phase-1 route. The source-bound bridges only synchronize the two proof routes. This theorem records that the analytic obligations and the two affected gate unitarity flags still have not been promoted. Guseynov-Huang-Liu 2025, Lemma 3, Eq. (20), and Eq. angles for Ry, arXiv:2506.20478.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:4610. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.249●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSharedRoute_flags_false
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract p row
            sparse).nonzeroNormalizer.proved =
      false ∧
    (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract p row
              sparse).divisionSemantics.proved =
        false ∧
      (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract p row
                sparse).coefficientBound.proved =
          false ∧
        (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract p
                  row sparse).absSquareSemantics.proved =
            false ∧
          (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract p
                    row sparse).sqrtComplementSemantics.proved =
              false ∧
            (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
                      p row sparse).arccosSemantics.proved =
                false ∧
              (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
                        p row sparse).twoByTwoUnitary.proved =
                  false ∧
                (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
                          p row sparse).coefficientDivision.proved =
                    false ∧
                  (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
                            p row sparse).normalizerBound.proved =
                      false ∧
                    (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
                              p row sparse).absSquareSemantics.proved =
                        false ∧
                      (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
                                p row
                                sparse).sqrtComplementSemantics.proved =
                          false ∧
                        (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
                                  p row sparse).twoByTwoUnitary.proved =
                            false ∧
                          (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                                    p row
                                    sparse).coefficientDivision.proved =
                              false ∧
                            (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                                      p row
                                      sparse).realArccosSemantics.proved =
                                false ∧
                              (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                                        p row
                                        sparse).halfAngleSemantics.proved =
                                  false ∧
                                (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                                          p row
                                          sparse).normalizerBound.proved =
                                    false ∧
                                  (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                                            p row
                                            sparse).twoByTwoUnitary.proved =
                                      false ∧
                                    (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_DT_S
                                              p).unitary.proved =
                                        false ∧
                                      (QuantumBlockEncoding.GHL2025.oneTermRobinGate_Ry_boundary
                                              p).unitary.proved =
                                        false
    theorem QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSharedRoute_flags_false
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
            p row
            sparse).nonzeroNormalizer.proved =
      false ∧
    (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
              p row
              sparse).divisionSemantics.proved =
        false ∧
      (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
                p row
                sparse).coefficientBound.proved =
          false ∧
        (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
                  p row
                  sparse).absSquareSemantics.proved =
            false ∧
          (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
                    p row
                    sparse).sqrtComplementSemantics.proved =
              false ∧
            (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
                      p row
                      sparse).arccosSemantics.proved =
                false ∧
              (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
                        p row
                        sparse).twoByTwoUnitary.proved =
                  false ∧
                (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
                          p row
                          sparse).coefficientDivision.proved =
                    false ∧
                  (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
                            p row
                            sparse).normalizerBound.proved =
                      false ∧
                    (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
                              p row
                              sparse).absSquareSemantics.proved =
                        false ∧
                      (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
                                p row
                                sparse).sqrtComplementSemantics.proved =
                          false ∧
                        (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
                                  p row
                                  sparse).twoByTwoUnitary.proved =
                            false ∧
                          (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                                    p row
                                    sparse).coefficientDivision.proved =
                              false ∧
                            (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                                      p
                                      row
                                      sparse).realArccosSemantics.proved =
                                false ∧
                              (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                                        p
                                        row
                                        sparse).halfAngleSemantics.proved =
                                  false ∧
                                (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                                          p
                                          row
                                          sparse).normalizerBound.proved =
                                    false ∧
                                  (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                                            p
                                            row
                                            sparse).twoByTwoUnitary.proved =
                                      false ∧
                                    (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_DT_S
                                              p).unitary.proved =
                                        false ∧
                                      (QuantumBlockEncoding.GHL2025.oneTermRobinGate_Ry_boundary
                                              p).unitary.proved =
                                        false
    Guard for the shared `N_D` Phase-1 route.

The source-bound bridges only synchronize the two proof routes.  This theorem
records that the analytic obligations and the two affected gate unitarity
flags still have not been promoted.
Guseynov-Huang-Liu 2025, Lemma 3, Eq. (20), and Eq. angles for Ry,
arXiv:2506.20478.
Theorem10.2.250
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “derivative normalizer nd shared route source bound and flags”; the hypotheses and conclusion in the code panel fix its exact scope. Combined Phase-1 guard for the shared 'N_D' route.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Combined Phase-1 guard for the shared 'N_D' route. This packages the source-bound bridges for 'O_DT^S' and 'Ry_boundary' together with the current false-flag state. It is bookkeeping only: the analytic division, bound, square-root, arccos, half-angle, and unitarity obligations are still unproved.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:4642. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.250●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSharedRoute_sourceBoundAndFlags
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  ((QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
            p row sparse).coefficient =
        (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
            p row sparse).sourceCoefficient ∧
      (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
              p row sparse).normalizerND =
          (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
              p row sparse).normalizerND ∧
        (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
              p row sparse).normalizerBound =
          (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
              p row sparse).coefficientBound) ∧
    ((QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
              p row sparse).coefficient =
          (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
              p row sparse).sourceCoefficient ∧
        (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                p row sparse).normalizerND =
            (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
                p row sparse).normalizerND ∧
          (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                p row sparse).normalizerBound =
            (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
                p row sparse).coefficientBound) ∧
      ((QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
                p row sparse).coefficient =
            (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                p row sparse).coefficient ∧
          (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
                  p row sparse).normalizerND =
              (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                  p row sparse).normalizerND ∧
            (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
                  p row sparse).normalizerBound =
              (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                  p row sparse).normalizerBound) ∧
        (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract p
                  row sparse).nonzeroNormalizer.proved =
            false ∧
          (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract p
                    row sparse).divisionSemantics.proved =
              false ∧
            (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
                      p row sparse).coefficientBound.proved =
                false ∧
              (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
                        p row sparse).absSquareSemantics.proved =
                  false ∧
                (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
                          p row sparse).sqrtComplementSemantics.proved =
                    false ∧
                  (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
                            p row sparse).arccosSemantics.proved =
                      false ∧
                    (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
                              p row sparse).twoByTwoUnitary.proved =
                        false ∧
                      (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
                                p row
                                sparse).coefficientDivision.proved =
                          false ∧
                        (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
                                  p row sparse).normalizerBound.proved =
                            false ∧
                          (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
                                    p row
                                    sparse).absSquareSemantics.proved =
                              false ∧
                            (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
                                      p row
                                      sparse).sqrtComplementSemantics.proved =
                                false ∧
                              (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
                                        p row
                                        sparse).twoByTwoUnitary.proved =
                                  false ∧
                                (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                                          p row
                                          sparse).coefficientDivision.proved =
                                    false ∧
                                  (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                                            p row
                                            sparse).realArccosSemantics.proved =
                                      false ∧
                                    (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                                              p row
                                              sparse).halfAngleSemantics.proved =
                                        false ∧
                                      (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                                                p row
                                                sparse).normalizerBound.proved =
                                          false ∧
                                        (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                                                  p row
                                                  sparse).twoByTwoUnitary.proved =
                                            false ∧
                                          (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_DT_S
                                                    p).unitary.proved =
                                              false ∧
                                            (QuantumBlockEncoding.GHL2025.oneTermRobinGate_Ry_boundary
                                                    p).unitary.proved =
                                              false
    theorem QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSharedRoute_sourceBoundAndFlags
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (row sparse : ℕ) :
  ((QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
            p row sparse).coefficient =
        (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
            p row
            sparse).sourceCoefficient ∧
      (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
              p row sparse).normalizerND =
          (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
              p row sparse).normalizerND ∧
        (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
              p row
              sparse).normalizerBound =
          (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
              p row
              sparse).coefficientBound) ∧
    ((QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
              p row sparse).coefficient =
          (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
              p row
              sparse).sourceCoefficient ∧
        (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                p row
                sparse).normalizerND =
            (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
                p row
                sparse).normalizerND ∧
          (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                p row
                sparse).normalizerBound =
            (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDSourceBound
                p row
                sparse).coefficientBound) ∧
      ((QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
                p row
                sparse).coefficient =
            (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                p row
                sparse).coefficient ∧
          (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
                  p row
                  sparse).normalizerND =
              (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                  p row
                  sparse).normalizerND ∧
            (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
                  p row
                  sparse).normalizerBound =
              (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                  p row
                  sparse).normalizerBound) ∧
        (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
                  p row
                  sparse).nonzeroNormalizer.proved =
            false ∧
          (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
                    p row
                    sparse).divisionSemantics.proved =
              false ∧
            (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
                      p row
                      sparse).coefficientBound.proved =
                false ∧
              (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
                        p row
                        sparse).absSquareSemantics.proved =
                  false ∧
                (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
                          p row
                          sparse).sqrtComplementSemantics.proved =
                    false ∧
                  (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
                            p row
                            sparse).arccosSemantics.proved =
                      false ∧
                    (QuantumBlockEncoding.GHL2025.derivativeNormalizerNDContract
                              p row
                              sparse).twoByTwoUnitary.proved =
                        false ∧
                      (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
                                p row
                                sparse).coefficientDivision.proved =
                          false ∧
                        (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
                                  p row
                                  sparse).normalizerBound.proved =
                            false ∧
                          (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
                                    p row
                                    sparse).absSquareSemantics.proved =
                              false ∧
                            (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
                                      p
                                      row
                                      sparse).sqrtComplementSemantics.proved =
                                false ∧
                              (QuantumBlockEncoding.GHL2025.sparseAmplitudeOracleDTCoefficientNormalizerProofRoute
                                        p
                                        row
                                        sparse).twoByTwoUnitary.proved =
                                  false ∧
                                (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                                          p
                                          row
                                          sparse).coefficientDivision.proved =
                                    false ∧
                                  (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                                            p
                                            row
                                            sparse).realArccosSemantics.proved =
                                      false ∧
                                    (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                                              p
                                              row
                                              sparse).halfAngleSemantics.proved =
                                        false ∧
                                      (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                                                p
                                                row
                                                sparse).normalizerBound.proved =
                                          false ∧
                                        (QuantumBlockEncoding.GHL2025.boundaryRotationAngleNormalizerProofRoute
                                                  p
                                                  row
                                                  sparse).twoByTwoUnitary.proved =
                                            false ∧
                                          (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_DT_S
                                                    p).unitary.proved =
                                              false ∧
                                            (QuantumBlockEncoding.GHL2025.oneTermRobinGate_Ry_boundary
                                                    p).unitary.proved =
                                              false
    Combined Phase-1 guard for the shared `N_D` route.

This packages the source-bound bridges for `O_DT^S` and `Ry_boundary` together
with the current false-flag state.  It is bookkeeping only: the analytic
division, bound, square-root, arccos, half-angle, and unitarity obligations are
still unproved.
Definition10.2.251
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access matrix”. Interim O_D^BS column-map helper, not the faithful Lemma 1 paper oracle.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Interim O_D^BS column-map helper, not the faithful Lemma 1 paper oracle. It maps |s⟩|i⟩ → |s⟩|col(s,i)⟩ by replacing the system register bits. Bits outside the system register are preserved. The paper contract |0>^(n-l)|s>^l|i>^n -> |r_si>^n|i>^n is recorded separately in 'defaultBandedSparseAccessPaperContract'; do not use this helper as the unitarity or block-extraction target for the paper oracle. main.tex:784-801 -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:4695. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.251●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessMatrix
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.Matrix
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
    QuantumBlockEncoding.Coeff
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessMatrix
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.Matrix
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p))
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p))
    QuantumBlockEncoding.Coeff
    Interim O_D^BS column-map helper, not the faithful Lemma 1 paper oracle.
It maps |s⟩|i⟩ → |s⟩|col(s,i)⟩ by replacing the system register bits.
Bits outside the system register are preserved.  The paper contract
|0>^(n-l)|s>^l|i>^n -> |r_si>^n|i>^n is recorded separately in
`defaultBandedSparseAccessPaperContract`; do not use this helper as the
unitarity or block-extraction target for the paper oracle.
main.tex:784-801 -
Definition10.2.252
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “one term robin gate o d bs”. Gate record for the faithful Lemma 1 O_D^BS paper-image matrix skeleton.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Gate record for the faithful Lemma 1 O_D^BS paper-image matrix skeleton. The matrix uses 'bandedSparseAccessPaperMatrix', which preserves the row register and writes 'r_si' into the padded sparse-address register. Unitarity, forward correctness, and block extraction remain unproved obligations. Guseynov-Huang-Liu 2025, Lemma 1, arXiv:2506.20478.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:4718. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.252●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GateMatrix QuantumBlockEncoding.Coeff
    (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)
    def QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GateMatrix
    QuantumBlockEncoding.Coeff
    (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
      p)
    Gate record for the faithful Lemma 1 O_D^BS paper-image matrix skeleton.

The matrix uses `bandedSparseAccessPaperMatrix`, which preserves the row
register and writes `r_si` into the padded sparse-address register.  Unitarity,
forward correctness, and block extraction remain unproved obligations.
Guseynov-Huang-Liu 2025, Lemma 1, arXiv:2506.20478.
Theorem10.2.253
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin gate o d bs image fin eq one”; the hypotheses and conclusion in the code panel fix its exact scope. Active forward 'O_D^BS' gate entry at the finite paper image.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Active forward 'O_D^BS' gate entry at the finite paper image.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:4728. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.253●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_imageFin_eq_one
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p ↑j <
      1 <<< p.n) :
  (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS p).matrix
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageFin p j
        haddr)
      j =
    QuantumBlockEncoding.Coeff.rat 1
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_imageFin_eq_one
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
        p ↑j <
      1 <<< p.n) :
  (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
          p).matrix
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageFin
        p j haddr)
      j =
    QuantumBlockEncoding.Coeff.rat 1
    Active forward `O_D^BS` gate entry at the finite paper image.
Theorem10.2.254
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin gate o d bs contract drift column 8 n 3”; the hypotheses and conclusion in the code panel fix its exact scope. Concrete contract-drift guard separating the active Lemma 1 paper-image matrix from the legacy sparse-column helper.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Concrete contract-drift guard separating the active Lemma 1 paper-image matrix from the legacy sparse-column helper. For the one-term parameters 'n = 3', 'kappa = 7', source column '8' is sent by the paper-image skeleton to row '40'. The active 'O_D^BS' gate therefore has entry '(40, 8) = 1' and no entry at '(4, 8)', while the legacy helper still has its historical row-'4' entry. This is only a regression guard; it does not promote unitarity, cleanup, or block extraction.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:4748. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.254●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_contractDrift_column8_n3 :
  let p :=
    { n := 3, kappa := 7, functionPieces := 1,
      polynomialDegreeCost := 1 };
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p 8 = 40 ∧
    (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS p).matrix
          ⟨40, ⋯⟩ ⟨8, ⋯⟩ =
        QuantumBlockEncoding.Coeff.rat 1 ∧
      (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS p).matrix
            ⟨4, ⋯⟩ ⟨8, ⋯⟩ =
          QuantumBlockEncoding.Coeff.rat 0 ∧
        QuantumBlockEncoding.GHL2025.bandedSparseAccessMatrix p ⟨4, ⋯⟩
              ⟨8, ⋯⟩ =
            QuantumBlockEncoding.Coeff.rat 1 ∧
          (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
                  p).unitary.proved =
            false
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_contractDrift_column8_n3 :
  let p :=
    { n := 3, kappa := 7,
      functionPieces := 1,
      polynomialDegreeCost := 1 };
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
        p 8 =
      40 ∧
    (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
              p).matrix
          ⟨40, ⋯⟩ ⟨8, ⋯⟩ =
        QuantumBlockEncoding.Coeff.rat 1 ∧
      (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
                p).matrix
            ⟨4, ⋯⟩ ⟨8, ⋯⟩ =
          QuantumBlockEncoding.Coeff.rat
            0 ∧
        QuantumBlockEncoding.GHL2025.bandedSparseAccessMatrix
              p ⟨4, ⋯⟩ ⟨8, ⋯⟩ =
            QuantumBlockEncoding.Coeff.rat
              1 ∧
          (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
                  p).unitary.proved =
            false
    Concrete contract-drift guard separating the active Lemma 1 paper-image matrix
from the legacy sparse-column helper.

For the one-term parameters `n = 3`, `kappa = 7`, source column `8` is sent by
the paper-image skeleton to row `40`.  The active `O_D^BS` gate therefore has
entry `(40, 8) = 1` and no entry at `(4, 8)`, while the legacy helper still has
its historical row-`4` entry.  This is only a regression guard; it does not
promote unitarity, cleanup, or block extraction.
Theorem10.2.255
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin gate o d bs boundary unused sparse collision n 3”; the hypotheses and conclusion in the code panel fix its exact scope. Concrete rejected-model collision for the old row-dependent 'O_D^BS' address.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Concrete rejected-model collision for the old row-dependent 'O_D^BS' address. For the one-term parameters 'n = 3', 'kappa = 7', boundary row '0' has only three nonzero Robin stencil entries. The old row-dependent address folded sparse index '3' back to the row address, colliding with sparse index '0'. The active global-slot paper image separates these columns; this theorem is retained only as regression memory for the rejected address model.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:4770. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.255●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_boundaryUnusedSparseCollision_n3 :
  have p :=
    { n := 3, kappa := 7, functionPieces := 1,
      polynomialDegreeCost := 1 };
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput p 0 =
      true ∧
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput p
          48 =
        true ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
              0).rowValue =
          0 ∧
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
                48).rowValue =
            0 ∧
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
                  p 0).sparseIndexValue =
              0 ∧
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
                    p 48).sparseIndexValue =
                3 ∧
              QuantumBlockEncoding.GHL2025.bandedSparseAccessRowDependentPaperAddress
                    p 0 =
                  0 ∧
                QuantumBlockEncoding.GHL2025.bandedSparseAccessRowDependentPaperAddress
                      p 48 =
                    0 ∧
                  QuantumBlockEncoding.GHL2025.bandedSparseAccessRowDependentPaperImage
                        p 0 =
                      QuantumBlockEncoding.GHL2025.bandedSparseAccessRowDependentPaperImage
                        p 48 ∧
                    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
                          p 0 =
                        6 ∧
                      QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
                            p 48 =
                          1 ∧
                        QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
                              p 0 ≠
                            QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
                              p 48 ∧
                          0 ≠ 48
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_boundaryUnusedSparseCollision_n3 :
  have p :=
    { n := 3, kappa := 7,
      functionPieces := 1,
      polynomialDegreeCost := 1 };
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput
        p 0 =
      true ∧
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput
          p 48 =
        true ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
              p 0).rowValue =
          0 ∧
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
                p 48).rowValue =
            0 ∧
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
                  p 0).sparseIndexValue =
              0 ∧
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
                    p
                    48).sparseIndexValue =
                3 ∧
              QuantumBlockEncoding.GHL2025.bandedSparseAccessRowDependentPaperAddress
                    p 0 =
                  0 ∧
                QuantumBlockEncoding.GHL2025.bandedSparseAccessRowDependentPaperAddress
                      p 48 =
                    0 ∧
                  QuantumBlockEncoding.GHL2025.bandedSparseAccessRowDependentPaperImage
                        p 0 =
                      QuantumBlockEncoding.GHL2025.bandedSparseAccessRowDependentPaperImage
                        p 48 ∧
                    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
                          p 0 =
                        6 ∧
                      QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
                            p 48 =
                          1 ∧
                        QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
                              p 0 ≠
                            QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
                              p 48 ∧
                          0 ≠ 48
    Concrete rejected-model collision for the old row-dependent `O_D^BS` address.

For the one-term parameters `n = 3`, `kappa = 7`, boundary row `0` has only
three nonzero Robin stencil entries.  The old row-dependent address folded
sparse index `3` back to the row address, colliding with sparse index `0`.
The active global-slot paper image separates these columns; this theorem is
retained only as regression memory for the rejected address model.
Theorem10.2.256
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin gate o d bs global sparse boundary no collision n 3”; the hypotheses and conclusion in the code panel fix its exact scope. Concrete regression that the corrected active global-slot image separates the old boundary unused-sparse collision columns.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Concrete regression that the corrected active global-slot image separates the old boundary unused-sparse collision columns. No semantic proof flag is promoted: this only checks the active image entries for the two concrete clean columns.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:4796. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.256●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_globalSparseBoundaryNoCollision_n3 :
  let p :=
    { n := 3, kappa := 7, functionPieces := 1,
      polynomialDegreeCost := 1 };
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput p 0 =
      true ∧
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput p
          48 =
        true ∧
      QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p 0 =
          6 ∧
        QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p
              48 =
            1 ∧
          QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p
                0 =
              96 ∧
            QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p
                  48 =
                16 ∧
              QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
                    p 0 ≠
                  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
                    p 48 ∧
                (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
                          p).matrix
                      ⟨96, ⋯⟩ ⟨0, ⋯⟩ =
                    QuantumBlockEncoding.Coeff.rat 1 ∧
                  (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
                            p).matrix
                        ⟨16, ⋯⟩ ⟨48, ⋯⟩ =
                      QuantumBlockEncoding.Coeff.rat 1 ∧
                    (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
                            p).unitary.proved =
                      false
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_globalSparseBoundaryNoCollision_n3 :
  let p :=
    { n := 3, kappa := 7,
      functionPieces := 1,
      polynomialDegreeCost := 1 };
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput
        p 0 =
      true ∧
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput
          p 48 =
        true ∧
      QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
            p 0 =
          6 ∧
        QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
              p 48 =
            1 ∧
          QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
                p 0 =
              96 ∧
            QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
                  p 48 =
                16 ∧
              QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
                    p 0 ≠
                  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
                    p 48 ∧
                (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
                          p).matrix
                      ⟨96, ⋯⟩ ⟨0, ⋯⟩ =
                    QuantumBlockEncoding.Coeff.rat
                      1 ∧
                  (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
                            p).matrix
                        ⟨16, ⋯⟩ ⟨48, ⋯⟩ =
                      QuantumBlockEncoding.Coeff.rat
                        1 ∧
                    (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
                            p).unitary.proved =
                      false
    Concrete regression that the corrected active global-slot image separates the
old boundary unused-sparse collision columns.

No semantic proof flag is promoted: this only checks the active image entries
for the two concrete clean columns.
Theorem10.2.257
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper global slot source boundary columns n 3”; the hypotheses and conclusion in the code panel fix its exact scope. The old boundary collision columns are both in the faithful global-slot source domain even though one of them is outside the rejected row-dependent nonzero-branch classifier.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The old boundary collision columns are both in the faithful global-slot source domain even though one of them is outside the rejected row-dependent nonzero-branch classifier. This is the regression that prevents future lower packets from treating 'bandedSparseAccessPaperValidCleanSource' as the active source predicate.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:4821. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.257●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource_boundaryColumns_n3 :
  have p :=
    { n := 3, kappa := 7, functionPieces := 1,
      polynomialDegreeCost := 1 };
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource p
        0 =
      true ∧
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
          p 48 =
        true ∧
      QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource
            p 48 =
          false ∧
        QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p 0 ≠
            QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p
              48 ∧
          (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
                  p).unitary.proved =
            false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource_boundaryColumns_n3 :
  have p :=
    { n := 3, kappa := 7,
      functionPieces := 1,
      polynomialDegreeCost := 1 };
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p 0 =
      true ∧
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
          p 48 =
        true ∧
      QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource
            p 48 =
          false ∧
        QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
              p 0 ≠
            QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
              p 48 ∧
          (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
                  p).unitary.proved =
            false
    The old boundary collision columns are both in the faithful global-slot source
domain even though one of them is outside the rejected row-dependent
nonzero-branch classifier.

This is the regression that prevents future lower packets from treating
`bandedSparseAccessPaperValidCleanSource` as the active source predicate.
Theorem10.2.258
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper global slot source encoded out of range n 3”; the hypotheses and conclusion in the code panel fix its exact scope. Encoded sparse value '7' is the first out-of-range slot for the one-term 'kappa = 7' contract.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Encoded sparse value '7' is the first out-of-range slot for the one-term 'kappa = 7' contract. It is clean in the padded O_D register but not in the faithful global-slot source domain.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:4836. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.258●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource_encodedOutOfRange_n3 :
  have p :=
    { n := 3, kappa := 7, functionPieces := 1,
      polynomialDegreeCost := 1 };
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput p 112 =
      true ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
            112).sparseIndexValue =
        7 ∧
      QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSparseIndexInKappa
            p 112 =
          false ∧
        QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
              p 112 =
            false ∧
          (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
                  p).unitary.proved =
            false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource_encodedOutOfRange_n3 :
  have p :=
    { n := 3, kappa := 7,
      functionPieces := 1,
      polynomialDegreeCost := 1 };
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput
        p 112 =
      true ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
            p 112).sparseIndexValue =
        7 ∧
      QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSparseIndexInKappa
            p 112 =
          false ∧
        QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
              p 112 =
            false ∧
          (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
                  p).unitary.proved =
            false
    Encoded sparse value `7` is the first out-of-range slot for the one-term
`kappa = 7` contract.  It is clean in the padded O_D register but not in the
faithful global-slot source domain.
Definition10.2.259
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “robin function value”. Symbolic function value at grid point j.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Symbolic function value at grid point j. Returns Coeff.symbol "f_x_j" for each grid index. The paper's O_f (Theorem amplitude-oracle for piece-wise polynomial function, main.tex:870-910) encodes f(x_j)/N_f; the 1/N_f factor is absorbed into the normalizer α = N_D · N_f · κ. main.tex:870-910 -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:4853. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.259●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.robinFunctionValue (n i : ℕ) :
  QuantumBlockEncoding.Coeff
    def QuantumBlockEncoding.GHL2025.robinFunctionValue
  (n i : ℕ) : QuantumBlockEncoding.Coeff
    Symbolic function value at grid point j.
Returns Coeff.symbol "f_x_j" for each grid index.
The paper's O_f (Theorem amplitude-oracle for piece-wise polynomial function,
main.tex:870-910) encodes f(x_j)/N_f;
the 1/N_f factor is absorbed into the normalizer α = N_D · N_f · κ.
main.tex:870-910 -
Definition10.2.260
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “function oracle paper registers”. A proposition-valued field is a requirement until a constructor supplies it. Register values used by the paper-level function oracle 'O_f' contract.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Register values used by the paper-level function oracle 'O_f' contract. The compound-index convention stores the system row in bits '[1, 1+n)' and stores the 'm_f' function-oracle workspace immediately above the indicator bit, starting at 'robinIndicatorBitPosition p + 1'. This record is a source-contract skeleton for the paper's clean-workspace equation; it does not assert the amplitude relation or workspace cleanup. Guseynov-Huang-Liu 2025, function-oracle construction, arXiv:2506.20478.

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:4865. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.260●1 definition
  • structure(4 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.FunctionOraclePaperRegisters : Type
    structure QuantumBlockEncoding.GHL2025.FunctionOraclePaperRegisters :
  Type
    Register values used by the paper-level function oracle `O_f` contract.

The compound-index convention stores the system row in bits `[1, 1+n)` and
stores the `m_f` function-oracle workspace immediately above the indicator bit,
starting at `robinIndicatorBitPosition p + 1`.  This record is a source-contract
skeleton for the paper's clean-workspace equation; it does not assert the
amplitude relation or workspace cleanup.
Guseynov-Huang-Liu 2025, function-oracle construction, arXiv:2506.20478.

    Fields

    systemValue : ℕ
    mfWorkspaceValue : ℕ
    nonMFValue : ℕ
    cleanWorkspace : Bool
Definition10.2.261
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “function oracle paper registers”. Extract the system register and the 'm_f' function workspace from a compound basis index for the 'O_f' source contract.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Extract the system register and the 'm_f' function workspace from a compound basis index for the 'O_f' source contract. The 'nonMFValue' field is the input index with the 'm_f' workspace bits cleared. For clean-workspace columns this is the clean-branch basis index appearing in the paper equation.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:4880. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.261●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.functionOraclePaperRegisters
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  QuantumBlockEncoding.GHL2025.FunctionOraclePaperRegisters
    def QuantumBlockEncoding.GHL2025.functionOraclePaperRegisters
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  QuantumBlockEncoding.GHL2025.FunctionOraclePaperRegisters
    Extract the system register and the `m_f` function workspace from a compound
basis index for the `O_f` source contract.

The `nonMFValue` field is the input index with the `m_f` workspace bits cleared.
For clean-workspace columns this is the clean-branch basis index appearing in
the paper equation.
Definition10.2.262
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “function oracle normalized value”. Symbolic normalized clean-branch amplitude for the paper's function oracle.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Symbolic normalized clean-branch amplitude for the paper's function oracle. The reciprocal symbol records the intended factor '1 / N_f' without proving that 'N_f' is nonzero or that the amplitude is bounded.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:4901. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.262●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.functionOracleNormalizedValue
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (i : ℕ) :
  QuantumBlockEncoding.Coeff
    def QuantumBlockEncoding.GHL2025.functionOracleNormalizedValue
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (i : ℕ) : QuantumBlockEncoding.Coeff
    Symbolic normalized clean-branch amplitude for the paper's function oracle.

The reciprocal symbol records the intended factor `1 / N_f` without proving
that `N_f` is nonzero or that the amplitude is bounded.
Definition10.2.263
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “function oracle paper image”. A proposition-valued field is a requirement until a constructor supplies it. Paper-image source contract for one column of the function oracle 'O_f'.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Paper-image source contract for one column of the function oracle 'O_f'. The clean branch records the displayed paper component '(f(x_i)/N_f)|0>^mf|i>'. The orthogonal component and all analytic side conditions are tracked as false obligations; this record is not a matrix proof and does not promote the current diagonal helper to a faithful oracle.

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:4912. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.263●1 definition
  • structure(14 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.FunctionOraclePaperImage : Type
    structure QuantumBlockEncoding.GHL2025.FunctionOraclePaperImage :
  Type
    Paper-image source contract for one column of the function oracle `O_f`.

The clean branch records the displayed paper component
`(f(x_i)/N_f)|0>^mf|i>`.  The orthogonal component and all analytic side
conditions are tracked as false obligations; this record is not a matrix proof
and does not promote the current diagonal helper to a faithful oracle.

    Fields

    sourceAnchor : String
    inputRegisters : QuantumBlockEncoding.GHL2025.FunctionOraclePaperRegisters
    cleanBranchBasisIndex : ℕ
    cleanBranchSystemValue : ℕ
    cleanBranchWorkspaceValue : ℕ
    cleanBranchAmplitude : QuantumBlockEncoding.Coeff
    orthogonalComponent : String
    systemPreserved : Bool
    cleanWorkspaceBranch : Bool
    normalizedAmplitudeCorrect : QuantumBlockEncoding.GHL2025.ObligationRecord
    orthogonalComponentCorrect : QuantumBlockEncoding.GHL2025.ObligationRecord
    normalizerBound : QuantumBlockEncoding.GHL2025.ObligationRecord
    unitaryCompletion : QuantumBlockEncoding.GHL2025.ObligationRecord
    diagonalHelperIsolation : QuantumBlockEncoding.GHL2025.ObligationRecord
Definition10.2.264
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “function oracle paper image”. Build the paper-level 'O_f' image contract for one compound basis column.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Build the paper-level 'O_f' image contract for one compound basis column. This captures the register-level target '|0>^mf|i> ↦ (f(x_i)/N_f)|0>^mf|i> + |orth_f(i)>' as data and keeps every unproved semantic claim explicit.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:4936. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.264●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.functionOraclePaperImage
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  QuantumBlockEncoding.GHL2025.FunctionOraclePaperImage
    def QuantumBlockEncoding.GHL2025.functionOraclePaperImage
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  QuantumBlockEncoding.GHL2025.FunctionOraclePaperImage
    Build the paper-level `O_f` image contract for one compound basis column.

This captures the register-level target
`|0>^mf|i> ↦ (f(x_i)/N_f)|0>^mf|i> + |orth_f(i)>` as data and keeps every
unproved semantic claim explicit.
Theorem10.2.265
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “function oracle paper image input registers eq”; the hypotheses and conclusion in the code panel fix its exact scope. Bridge lemma: the 'O_f' paper image uses the shared register extractor.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Bridge lemma: the 'O_f' paper image uses the shared register extractor.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:4977. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.265●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.functionOraclePaperImage_inputRegisters_eq
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.functionOraclePaperImage p
        j).inputRegisters =
    QuantumBlockEncoding.GHL2025.functionOraclePaperRegisters p j
    theorem QuantumBlockEncoding.GHL2025.functionOraclePaperImage_inputRegisters_eq
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.functionOraclePaperImage
        p j).inputRegisters =
    QuantumBlockEncoding.GHL2025.functionOraclePaperRegisters
      p j
    Bridge lemma: the `O_f` paper image uses the shared register extractor.
Theorem10.2.266
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “function oracle paper image clean branch basis index eq”; the hypotheses and conclusion in the code panel fix its exact scope. Bridge lemma: the clean 'O_f' branch clears only the 'm_f' workspace bits.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Bridge lemma: the clean 'O_f' branch clears only the 'm_f' workspace bits.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:4983. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.266●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.functionOraclePaperImage_cleanBranchBasisIndex_eq
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.functionOraclePaperImage p
        j).cleanBranchBasisIndex =
    (QuantumBlockEncoding.GHL2025.functionOraclePaperRegisters p
        j).nonMFValue
    theorem QuantumBlockEncoding.GHL2025.functionOraclePaperImage_cleanBranchBasisIndex_eq
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.functionOraclePaperImage
        p j).cleanBranchBasisIndex =
    (QuantumBlockEncoding.GHL2025.functionOraclePaperRegisters
        p j).nonMFValue
    Bridge lemma: the clean `O_f` branch clears only the `m_f` workspace bits.
Theorem10.2.267
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “function oracle paper image clean branch system value eq”; the hypotheses and conclusion in the code panel fix its exact scope. Bridge lemma: the clean 'O_f' branch preserves the extracted system value.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Bridge lemma: the clean 'O_f' branch preserves the extracted system value.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:4989. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.267●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.functionOraclePaperImage_cleanBranchSystemValue_eq
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.functionOraclePaperImage p
        j).cleanBranchSystemValue =
    (QuantumBlockEncoding.GHL2025.functionOraclePaperRegisters p
        j).systemValue
    theorem QuantumBlockEncoding.GHL2025.functionOraclePaperImage_cleanBranchSystemValue_eq
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.functionOraclePaperImage
        p j).cleanBranchSystemValue =
    (QuantumBlockEncoding.GHL2025.functionOraclePaperRegisters
        p j).systemValue
    Bridge lemma: the clean `O_f` branch preserves the extracted system value.
Theorem10.2.268
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “function oracle paper image clean branch workspace value eq”; the hypotheses and conclusion in the code panel fix its exact scope. Bridge lemma: the clean 'O_f' branch has zero 'm_f' workspace value.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Bridge lemma: the clean 'O_f' branch has zero 'm_f' workspace value.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:4995. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.268●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.functionOraclePaperImage_cleanBranchWorkspaceValue_eq
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.functionOraclePaperImage p
        j).cleanBranchWorkspaceValue =
    0
    theorem QuantumBlockEncoding.GHL2025.functionOraclePaperImage_cleanBranchWorkspaceValue_eq
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.functionOraclePaperImage
        p j).cleanBranchWorkspaceValue =
    0
    Bridge lemma: the clean `O_f` branch has zero `m_f` workspace value.
Theorem10.2.269
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “function oracle paper image clean branch amplitude eq”; the hypotheses and conclusion in the code panel fix its exact scope. Bridge lemma: the clean 'O_f' branch amplitude is the normalized function value at the system value extracted from the same column.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Bridge lemma: the clean 'O_f' branch amplitude is the normalized function value at the system value extracted from the same column.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5003. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.269●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.functionOraclePaperImage_cleanBranchAmplitude_eq
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.functionOraclePaperImage p
        j).cleanBranchAmplitude =
    QuantumBlockEncoding.GHL2025.functionOracleNormalizedValue p
      (QuantumBlockEncoding.GHL2025.functionOraclePaperRegisters p
          j).systemValue
    theorem QuantumBlockEncoding.GHL2025.functionOraclePaperImage_cleanBranchAmplitude_eq
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.functionOraclePaperImage
        p j).cleanBranchAmplitude =
    QuantumBlockEncoding.GHL2025.functionOracleNormalizedValue
      p
      (QuantumBlockEncoding.GHL2025.functionOraclePaperRegisters
          p j).systemValue
    Bridge lemma: the clean `O_f` branch amplitude is the normalized function value
at the system value extracted from the same column.
Theorem10.2.270
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “function oracle paper image clean workspace branch eq”; the hypotheses and conclusion in the code panel fix its exact scope. Bridge lemma: the clean-workspace branch flag is inherited from the extractor.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Bridge lemma: the clean-workspace branch flag is inherited from the extractor.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5009. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.270●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.functionOraclePaperImage_cleanWorkspaceBranch_eq
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.functionOraclePaperImage p
        j).cleanWorkspaceBranch =
    (QuantumBlockEncoding.GHL2025.functionOraclePaperRegisters p
        j).cleanWorkspace
    theorem QuantumBlockEncoding.GHL2025.functionOraclePaperImage_cleanWorkspaceBranch_eq
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.functionOraclePaperImage
        p j).cleanWorkspaceBranch =
    (QuantumBlockEncoding.GHL2025.functionOraclePaperRegisters
        p j).cleanWorkspace
    Bridge lemma: the clean-workspace branch flag is inherited from the extractor.
Definition10.2.271
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “function oracle external amplitude source contract”. A proposition-valued field is a requirement until a constructor supplies it. External source transcript for the O_f amplitude-oracle theorem cited by GHL2025.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. External source transcript for the O_f amplitude-oracle theorem cited by GHL2025. This records the theorem and coordinate-oracle equation used as a source contract for the function oracle. It does not formalize the cited theorem and does not close the analytic facts needed for the Lean 'O_f' contract.

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:5022. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.271●1 definition
  • structure(15 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.FunctionOracleExternalAmplitudeSourceContract :
  Type
    structure QuantumBlockEncoding.GHL2025.FunctionOracleExternalAmplitudeSourceContract :
  Type
    External source transcript for the O_f amplitude-oracle theorem cited by
GHL2025.

This records the theorem and coordinate-oracle equation used as a source
contract for the function oracle.  It does not formalize the cited theorem and
does not close the analytic facts needed for the Lean `O_f` contract.

    Fields

    sourceAnchor : String
    theoremAnchor : String
    coordinateOracleAnchor : String
    citedSourceAnchor : String
    cleanBranchFormula : String
    normalizerNf : QuantumBlockEncoding.Coeff
    resourceClaim : QuantumBlockEncoding.GHL2025.ObligationRecord
    externalTheoremFormalized : QuantumBlockEncoding.GHL2025.ObligationRecord
    nonzeroNormalizer : QuantumBlockEncoding.GHL2025.ObligationRecord
    divisionSemantics : QuantumBlockEncoding.GHL2025.ObligationRecord
    theoremAmplitudeCorrect : QuantumBlockEncoding.GHL2025.ObligationRecord
    closesNormalizerBound : Bool
    closesOrthogonalCompletion : Bool
    closesUnitaryCompletion : Bool
    closesFunctionOracleContract : Bool
Definition10.2.272
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “function oracle external amplitude source contract”. Default source transcript for GHL2025's function-oracle dependency.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Default source transcript for GHL2025's function-oracle dependency. The GHL2025 theorem cites Guseynov--Liu 2024, arXiv:2411.01131, Theorem 5. The QBE status remains obligation-only: this declaration gives later proof packets a typed source anchor, not a proof of the theorem.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:5047. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.272●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract :
  QuantumBlockEncoding.GHL2025.FunctionOracleExternalAmplitudeSourceContract
    def QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract :
  QuantumBlockEncoding.GHL2025.FunctionOracleExternalAmplitudeSourceContract
    Default source transcript for GHL2025's function-oracle dependency.

The GHL2025 theorem cites Guseynov--Liu 2024, arXiv:2411.01131, Theorem 5.
The QBE status remains obligation-only: this declaration gives later proof
packets a typed source anchor, not a proof of the theorem.
Theorem10.2.273
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “function oracle external amplitude source contract source anchor”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5088. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.273●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract_sourceAnchor :
  QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.sourceAnchor =
    "GHL2025 Theorem 'Amplitude-oracle for piece-wise polynomial function' and Eq. 'coordinate oracle', arXiv:2506.20478; cited source arXiv:2411.01131"
    theorem QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract_sourceAnchor :
  QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.sourceAnchor =
    "GHL2025 Theorem 'Amplitude-oracle for piece-wise polynomial function' and Eq. 'coordinate oracle', arXiv:2506.20478; cited source arXiv:2411.01131"
Theorem10.2.274
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “function oracle external amplitude source contract flags false”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5092. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.274●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract_flags_false :
  QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.resourceClaim.proved =
      false ∧
    QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.externalTheoremFormalized.proved =
        false ∧
      QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.nonzeroNormalizer.proved =
          false ∧
        QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.divisionSemantics.proved =
            false ∧
          QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.theoremAmplitudeCorrect.proved =
              false ∧
            QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.closesNormalizerBound =
                false ∧
              QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.closesOrthogonalCompletion =
                  false ∧
                QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.closesUnitaryCompletion =
                    false ∧
                  QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.closesFunctionOracleContract =
                    false
    theorem QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract_flags_false :
  QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.resourceClaim.proved =
      false ∧
    QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.externalTheoremFormalized.proved =
        false ∧
      QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.nonzeroNormalizer.proved =
          false ∧
        QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.divisionSemantics.proved =
            false ∧
          QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.theoremAmplitudeCorrect.proved =
              false ∧
            QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.closesNormalizerBound =
                false ∧
              QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.closesOrthogonalCompletion =
                  false ∧
                QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.closesUnitaryCompletion =
                    false ∧
                  QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.closesFunctionOracleContract =
                    false
Definition10.2.275
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “function oracle amplitude proof route”. A proposition-valued field is a requirement until a constructor supplies it. Refined proof route for the 'of_nf_amplitude_route' block.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Refined proof route for the 'of_nf_amplitude_route' block. The route ties the paper's coordinate-oracle equation to the current Lean source-contract data: the symbolic function value, the 'N_f' normalizer symbol, the clean-branch amplitude in 'functionOraclePaperImage', and the theorem-level function-oracle obligation. It does not prove that 'N_f' is nonzero, that 'N_f_inv' is an inverse, that the normalizer bound holds, or that the orthogonal component gives a unitary completion.

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:5114. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.275●1 definition
  • structure(17 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.FunctionOracleAmplitudeProofRoute : Type
    structure QuantumBlockEncoding.GHL2025.FunctionOracleAmplitudeProofRoute :
  Type
    Refined proof route for the `of_nf_amplitude_route` block.

The route ties the paper's coordinate-oracle equation to the current Lean
source-contract data: the symbolic function value, the `N_f` normalizer symbol,
the clean-branch amplitude in `functionOraclePaperImage`, and the theorem-level
function-oracle obligation.  It does not prove that `N_f` is nonzero, that
`N_f_inv` is an inverse, that the normalizer bound holds, or that the
orthogonal component gives a unitary completion.

    Fields

    sourceAnchor : String
    systemValue : ℕ
    sourceFunctionValue : QuantumBlockEncoding.Coeff
    normalizerNf : QuantumBlockEncoding.Coeff
    normalizedAmplitude : QuantumBlockEncoding.Coeff
    normalizedAmplitudeFormula : String
    cleanBranchAmplitude : QuantumBlockEncoding.Coeff
    cleanBranchBasisIndex : ℕ
    cleanWorkspaceBranch : Bool
    theoremNormalizer : QuantumBlockEncoding.Coeff
    normalizedAmplitudeCorrect : QuantumBlockEncoding.GHL2025.ObligationRecord
    nonzeroNormalizer : QuantumBlockEncoding.GHL2025.ObligationRecord
    divisionSemantics : QuantumBlockEncoding.GHL2025.ObligationRecord
    normalizerBound : QuantumBlockEncoding.GHL2025.ObligationRecord
    orthogonalComponentCorrect : QuantumBlockEncoding.GHL2025.ObligationRecord
    unitaryCompletion : QuantumBlockEncoding.GHL2025.ObligationRecord
    theoremAmplitudeCorrect : QuantumBlockEncoding.GHL2025.ObligationRecord
Definition10.2.276
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “function oracle amplitude proof route”. Default O_f amplitude-route contract for one compound basis column.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Default O_f amplitude-route contract for one compound basis column. The route reuses 'functionOraclePaperImage'; it only packages the dependencies needed before any future proof of the clean-branch amplitude or theorem-level 'FunctionOracleContract.amplitudeCorrect' field.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:5141. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.276●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  QuantumBlockEncoding.GHL2025.FunctionOracleAmplitudeProofRoute
    def QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  QuantumBlockEncoding.GHL2025.FunctionOracleAmplitudeProofRoute
    Default O_f amplitude-route contract for one compound basis column.

The route reuses `functionOraclePaperImage`; it only packages the dependencies
needed before any future proof of the clean-branch amplitude or theorem-level
`FunctionOracleContract.amplitudeCorrect` field.
Theorem10.2.277
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “function oracle amplitude proof route source anchor”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5166. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.277●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute_sourceAnchor
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute p
        j).sourceAnchor =
    "GHL2025 Theorem 'Amplitude-oracle for piece-wise polynomial function' and Eq. 'coordinate oracle', arXiv:2506.20478; cited source arXiv:2411.01131"
    theorem QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute_sourceAnchor
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
        p j).sourceAnchor =
    "GHL2025 Theorem 'Amplitude-oracle for piece-wise polynomial function' and Eq. 'coordinate oracle', arXiv:2506.20478; cited source arXiv:2411.01131"
Theorem10.2.278
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “function oracle amplitude proof route source function value”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5171. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.278●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute_sourceFunctionValue
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute p
        j).sourceFunctionValue =
    QuantumBlockEncoding.GHL2025.robinFunctionValue p.n
      (QuantumBlockEncoding.GHL2025.functionOraclePaperRegisters p
          j).systemValue
    theorem QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute_sourceFunctionValue
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
        p j).sourceFunctionValue =
    QuantumBlockEncoding.GHL2025.robinFunctionValue
      p.n
      (QuantumBlockEncoding.GHL2025.functionOraclePaperRegisters
          p j).systemValue
Theorem10.2.279
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “function oracle amplitude proof route normalizer nf”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5176. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.279●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute_normalizerNf
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute p
        j).normalizerNf =
    QuantumBlockEncoding.Coeff.symbol "N_f"
    theorem QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute_normalizerNf
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
        p j).normalizerNf =
    QuantumBlockEncoding.Coeff.symbol
      "N_f"
Theorem10.2.280
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “function oracle amplitude proof route normalized amplitude”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5181. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.280●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute_normalizedAmplitude
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute p
        j).normalizedAmplitude =
    QuantumBlockEncoding.GHL2025.functionOracleNormalizedValue p
      (QuantumBlockEncoding.GHL2025.functionOraclePaperRegisters p
          j).systemValue
    theorem QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute_normalizedAmplitude
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
        p j).normalizedAmplitude =
    QuantumBlockEncoding.GHL2025.functionOracleNormalizedValue
      p
      (QuantumBlockEncoding.GHL2025.functionOraclePaperRegisters
          p j).systemValue
Theorem10.2.281
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “function oracle amplitude proof route paper image”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5186. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.281●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute_paperImage
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute p
          j).cleanBranchAmplitude =
      (QuantumBlockEncoding.GHL2025.functionOraclePaperImage p
          j).cleanBranchAmplitude ∧
    (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute p
            j).cleanBranchBasisIndex =
        (QuantumBlockEncoding.GHL2025.functionOraclePaperImage p
            j).cleanBranchBasisIndex ∧
      (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute p
            j).cleanWorkspaceBranch =
        (QuantumBlockEncoding.GHL2025.functionOraclePaperImage p
            j).cleanWorkspaceBranch
    theorem QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute_paperImage
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
          p j).cleanBranchAmplitude =
      (QuantumBlockEncoding.GHL2025.functionOraclePaperImage
          p j).cleanBranchAmplitude ∧
    (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
            p j).cleanBranchBasisIndex =
        (QuantumBlockEncoding.GHL2025.functionOraclePaperImage
            p j).cleanBranchBasisIndex ∧
      (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
            p j).cleanWorkspaceBranch =
        (QuantumBlockEncoding.GHL2025.functionOraclePaperImage
            p j).cleanWorkspaceBranch
Theorem10.2.282
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “function oracle amplitude proof route obligations reuse paper image”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5196. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.282●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute_obligations_reuse_paperImage
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute p
          j).normalizedAmplitudeCorrect =
      (QuantumBlockEncoding.GHL2025.functionOraclePaperImage p
          j).normalizedAmplitudeCorrect ∧
    (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute p
            j).normalizerBound =
        (QuantumBlockEncoding.GHL2025.functionOraclePaperImage p
            j).normalizerBound ∧
      (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute p
              j).orthogonalComponentCorrect =
          (QuantumBlockEncoding.GHL2025.functionOraclePaperImage p
              j).orthogonalComponentCorrect ∧
        (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
              p j).unitaryCompletion =
          (QuantumBlockEncoding.GHL2025.functionOraclePaperImage p
              j).unitaryCompletion
    theorem QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute_obligations_reuse_paperImage
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
          p
          j).normalizedAmplitudeCorrect =
      (QuantumBlockEncoding.GHL2025.functionOraclePaperImage
          p
          j).normalizedAmplitudeCorrect ∧
    (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
            p j).normalizerBound =
        (QuantumBlockEncoding.GHL2025.functionOraclePaperImage
            p j).normalizerBound ∧
      (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
              p
              j).orthogonalComponentCorrect =
          (QuantumBlockEncoding.GHL2025.functionOraclePaperImage
              p
              j).orthogonalComponentCorrect ∧
        (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
              p j).unitaryCompletion =
          (QuantumBlockEncoding.GHL2025.functionOraclePaperImage
              p j).unitaryCompletion
Theorem10.2.283
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “function oracle amplitude proof route external source contract”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5208. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.283●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute_externalSourceContract
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute p
          j).sourceAnchor =
      QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.sourceAnchor ∧
    (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute p
            j).normalizerNf =
        QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.normalizerNf ∧
      (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute p
              j).normalizedAmplitudeFormula =
          QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.cleanBranchFormula ∧
        (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                p j).nonzeroNormalizer =
            QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.nonzeroNormalizer ∧
          (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                  p j).divisionSemantics =
              QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.divisionSemantics ∧
            (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                  p j).theoremAmplitudeCorrect =
              QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.theoremAmplitudeCorrect
    theorem QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute_externalSourceContract
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
          p j).sourceAnchor =
      QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.sourceAnchor ∧
    (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
            p j).normalizerNf =
        QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.normalizerNf ∧
      (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
              p
              j).normalizedAmplitudeFormula =
          QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.cleanBranchFormula ∧
        (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                p j).nonzeroNormalizer =
            QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.nonzeroNormalizer ∧
          (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                  p j).divisionSemantics =
              QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.divisionSemantics ∧
            (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                  p
                  j).theoremAmplitudeCorrect =
              QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.theoremAmplitudeCorrect
Theorem10.2.284
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “function oracle amplitude proof route flags false”; the hypotheses and conclusion in the code panel fix its exact scope.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The source declaration has no docstring. The reader cue above is generated from its kind and name and does not replace the Lean signature.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5224. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.284●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute_flags_false
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute p
            j).normalizedAmplitudeCorrect.proved =
      false ∧
    (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute p
              j).nonzeroNormalizer.proved =
        false ∧
      (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute p
                j).divisionSemantics.proved =
          false ∧
        (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                  p j).normalizerBound.proved =
            false ∧
          (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                    p j).orthogonalComponentCorrect.proved =
              false ∧
            (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                      p j).unitaryCompletion.proved =
                false ∧
              (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                      p j).theoremAmplitudeCorrect.proved =
                false
    theorem QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute_flags_false
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
            p
            j).normalizedAmplitudeCorrect.proved =
      false ∧
    (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
              p
              j).nonzeroNormalizer.proved =
        false ∧
      (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                p
                j).divisionSemantics.proved =
          false ∧
        (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                  p
                  j).normalizerBound.proved =
            false ∧
          (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                    p
                    j).orthogonalComponentCorrect.proved =
              false ∧
            (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                      p
                      j).unitaryCompletion.proved =
                false ∧
              (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                      p
                      j).theoremAmplitudeCorrect.proved =
                false
Theorem10.2.285
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “function oracle amplitude proof route external source and flags”; the hypotheses and conclusion in the code panel fix its exact scope. Combined Phase-1 guard for the 'O_f' external-source route.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Combined Phase-1 guard for the 'O_f' external-source route. The bridge to the cited amplitude-oracle theorem and the false analytic flags are packaged together so later proof packets cannot use the source transcript as a proof of 'O_f' amplitude correctness or unitarity.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5242. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.285●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute_externalSourceAndFlags
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  ((QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute p
            j).sourceAnchor =
        QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.sourceAnchor ∧
      (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute p
              j).normalizerNf =
          QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.normalizerNf ∧
        (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                p j).normalizedAmplitudeFormula =
            QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.cleanBranchFormula ∧
          (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                  p j).nonzeroNormalizer =
              QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.nonzeroNormalizer ∧
            (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                    p j).divisionSemantics =
                QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.divisionSemantics ∧
              (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                    p j).theoremAmplitudeCorrect =
                QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.theoremAmplitudeCorrect) ∧
    (QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.resourceClaim.proved =
          false ∧
        QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.externalTheoremFormalized.proved =
            false ∧
          QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.nonzeroNormalizer.proved =
              false ∧
            QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.divisionSemantics.proved =
                false ∧
              QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.theoremAmplitudeCorrect.proved =
                  false ∧
                QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.closesNormalizerBound =
                    false ∧
                  QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.closesOrthogonalCompletion =
                      false ∧
                    QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.closesUnitaryCompletion =
                        false ∧
                      QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.closesFunctionOracleContract =
                        false) ∧
      (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute p
                j).normalizedAmplitudeCorrect.proved =
          false ∧
        (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                  p j).nonzeroNormalizer.proved =
            false ∧
          (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                    p j).divisionSemantics.proved =
              false ∧
            (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                      p j).normalizerBound.proved =
                false ∧
              (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                        p j).orthogonalComponentCorrect.proved =
                  false ∧
                (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                          p j).unitaryCompletion.proved =
                    false ∧
                  (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                          p j).theoremAmplitudeCorrect.proved =
                    false
    theorem QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute_externalSourceAndFlags
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  ((QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
            p j).sourceAnchor =
        QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.sourceAnchor ∧
      (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
              p j).normalizerNf =
          QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.normalizerNf ∧
        (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                p
                j).normalizedAmplitudeFormula =
            QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.cleanBranchFormula ∧
          (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                  p j).nonzeroNormalizer =
              QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.nonzeroNormalizer ∧
            (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                    p
                    j).divisionSemantics =
                QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.divisionSemantics ∧
              (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                    p
                    j).theoremAmplitudeCorrect =
                QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.theoremAmplitudeCorrect) ∧
    (QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.resourceClaim.proved =
          false ∧
        QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.externalTheoremFormalized.proved =
            false ∧
          QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.nonzeroNormalizer.proved =
              false ∧
            QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.divisionSemantics.proved =
                false ∧
              QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.theoremAmplitudeCorrect.proved =
                  false ∧
                QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.closesNormalizerBound =
                    false ∧
                  QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.closesOrthogonalCompletion =
                      false ∧
                    QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.closesUnitaryCompletion =
                        false ∧
                      QuantumBlockEncoding.GHL2025.functionOracleExternalAmplitudeSourceContract.closesFunctionOracleContract =
                        false) ∧
      (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                p
                j).normalizedAmplitudeCorrect.proved =
          false ∧
        (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                  p
                  j).nonzeroNormalizer.proved =
            false ∧
          (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                    p
                    j).divisionSemantics.proved =
              false ∧
            (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                      p
                      j).normalizerBound.proved =
                false ∧
              (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                        p
                        j).orthogonalComponentCorrect.proved =
                  false ∧
                (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                          p
                          j).unitaryCompletion.proved =
                    false ∧
                  (QuantumBlockEncoding.GHL2025.functionOracleAmplitudeProofRoute
                          p
                          j).theoremAmplitudeCorrect.proved =
                    false
    Combined Phase-1 guard for the `O_f` external-source route.

The bridge to the cited amplitude-oracle theorem and the false analytic flags
are packaged together so later proof packets cannot use the source transcript
as a proof of `O_f` amplitude correctness or unitarity.
Definition10.2.286
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “function oracle orthogonal entry”. Symbolic matrix entry for the unresolved orthogonal component of 'O_f'.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Symbolic matrix entry for the unresolved orthogonal component of 'O_f'. The paper only fixes the clean 'm_f' branch amplitude 'f(x_i) / N_f'; the remaining orthogonal completion is a unitarity obligation. This symbol records one placeholder entry for that unresolved completion without proving orthogonality, normalizer bounds, or unitarity.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:5285. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.286●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.functionOracleOrthogonalEntry
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (systemValue row col : ℕ) : QuantumBlockEncoding.Coeff
    def QuantumBlockEncoding.GHL2025.functionOracleOrthogonalEntry
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (systemValue row col : ℕ) :
  QuantumBlockEncoding.Coeff
    Symbolic matrix entry for the unresolved orthogonal component of `O_f`.

The paper only fixes the clean `m_f` branch amplitude
`f(x_i) / N_f`; the remaining orthogonal completion is a unitarity obligation.
This symbol records one placeholder entry for that unresolved completion without
proving orthogonality, normalizer bounds, or unitarity.
Definition10.2.287
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “function oracle paper matrix”. Faithful Phase 1 matrix skeleton for the paper-level function oracle 'O_f'.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Faithful Phase 1 matrix skeleton for the paper-level function oracle 'O_f'. For each clean-workspace input column, the clean 'm_f' branch entry is the normalized amplitude recorded by 'functionOraclePaperImage', namely 'f(x_i) / N_f' represented as 'functionOracleNormalizedValue'. Other clean-workspace output rows are zero, matching the paper statement that the unresolved component is orthogonal to the clean workspace branch. Non-clean-workspace rows carry symbolic completion entries. For non-clean input columns, the paper does not fix a branch equation, so this skeleton leaves all entries symbolic. The symbolic completion does not prove amplitude correctness, the 'N_f' bound, orthogonality, or unitarity; those obligations remain false in 'functionOraclePaperImage' and the gate record.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:5305. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.287●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.functionOraclePaperMatrix
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.Matrix
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
    QuantumBlockEncoding.Coeff
    def QuantumBlockEncoding.GHL2025.functionOraclePaperMatrix
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.Matrix
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p))
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p))
    QuantumBlockEncoding.Coeff
    Faithful Phase 1 matrix skeleton for the paper-level function oracle `O_f`.

For each clean-workspace input column, the clean `m_f` branch entry is the
normalized amplitude recorded by `functionOraclePaperImage`, namely
`f(x_i) / N_f` represented as `functionOracleNormalizedValue`.  Other
clean-workspace output rows are zero, matching the paper statement that the
unresolved component is orthogonal to the clean workspace branch.
Non-clean-workspace rows carry symbolic completion entries.  For non-clean
input columns, the paper does not fix a branch equation, so this skeleton leaves
all entries symbolic.

The symbolic completion does not prove amplitude correctness, the `N_f` bound,
orthogonality, or unitarity; those obligations remain false in
`functionOraclePaperImage` and the gate record.
Theorem10.2.288
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “function oracle paper matrix clean branch entry”; the hypotheses and conclusion in the code panel fix its exact scope. The 'O_f' paper matrix exposes the clean branch amplitude for clean input columns.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The 'O_f' paper matrix exposes the clean branch amplitude for clean input columns.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5320. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.288●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.functionOraclePaperMatrix_cleanBranch_entry
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (i j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (hClean :
    (QuantumBlockEncoding.GHL2025.functionOraclePaperImage p
          ↑j).cleanWorkspaceBranch =
      true)
  (h :
    ↑i =
      (QuantumBlockEncoding.GHL2025.functionOraclePaperImage p
          ↑j).cleanBranchBasisIndex) :
  QuantumBlockEncoding.GHL2025.functionOraclePaperMatrix p i j =
    (QuantumBlockEncoding.GHL2025.functionOraclePaperImage p
        ↑j).cleanBranchAmplitude
    theorem QuantumBlockEncoding.GHL2025.functionOraclePaperMatrix_cleanBranch_entry
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (i j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (hClean :
    (QuantumBlockEncoding.GHL2025.functionOraclePaperImage
          p ↑j).cleanWorkspaceBranch =
      true)
  (h :
    ↑i =
      (QuantumBlockEncoding.GHL2025.functionOraclePaperImage
          p ↑j).cleanBranchBasisIndex) :
  QuantumBlockEncoding.GHL2025.functionOraclePaperMatrix
      p i j =
    (QuantumBlockEncoding.GHL2025.functionOraclePaperImage
        p ↑j).cleanBranchAmplitude
    The `O_f` paper matrix exposes the clean branch amplitude for clean input columns.
Theorem10.2.289
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “function oracle paper matrix clean workspace off branch zero”; the hypotheses and conclusion in the code panel fix its exact scope. Other clean-workspace rows have zero 'O_f' orthogonal-completion entry.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Other clean-workspace rows have zero 'O_f' orthogonal-completion entry.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5330. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.289●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.functionOraclePaperMatrix_cleanWorkspace_offBranch_zero
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (i j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (hInputClean :
    (QuantumBlockEncoding.GHL2025.functionOraclePaperImage p
          ↑j).cleanWorkspaceBranch =
      true)
  (hBranch :
    ↑i ≠
      (QuantumBlockEncoding.GHL2025.functionOraclePaperImage p
          ↑j).cleanBranchBasisIndex)
  (hClean :
    (QuantumBlockEncoding.GHL2025.functionOraclePaperRegisters p
          ↑i).mfWorkspaceValue =
      0) :
  QuantumBlockEncoding.GHL2025.functionOraclePaperMatrix p i j =
    QuantumBlockEncoding.Coeff.rat 0
    theorem QuantumBlockEncoding.GHL2025.functionOraclePaperMatrix_cleanWorkspace_offBranch_zero
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (i j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (hInputClean :
    (QuantumBlockEncoding.GHL2025.functionOraclePaperImage
          p ↑j).cleanWorkspaceBranch =
      true)
  (hBranch :
    ↑i ≠
      (QuantumBlockEncoding.GHL2025.functionOraclePaperImage
          p ↑j).cleanBranchBasisIndex)
  (hClean :
    (QuantumBlockEncoding.GHL2025.functionOraclePaperRegisters
          p ↑i).mfWorkspaceValue =
      0) :
  QuantumBlockEncoding.GHL2025.functionOraclePaperMatrix
      p i j =
    QuantumBlockEncoding.Coeff.rat 0
    Other clean-workspace rows have zero `O_f` orthogonal-completion entry.
Theorem10.2.290
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “function oracle paper matrix non clean input entry”; the hypotheses and conclusion in the code panel fix its exact scope. Non-clean input columns are left in the symbolic 'O_f' completion branch.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Non-clean input columns are left in the symbolic 'O_f' completion branch.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5340. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.290●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.functionOraclePaperMatrix_nonCleanInput_entry
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (i j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (hInputNonClean :
    (QuantumBlockEncoding.GHL2025.functionOraclePaperImage p
          ↑j).cleanWorkspaceBranch =
      false) :
  QuantumBlockEncoding.GHL2025.functionOraclePaperMatrix p i j =
    QuantumBlockEncoding.GHL2025.functionOracleOrthogonalEntry p
      (QuantumBlockEncoding.GHL2025.functionOraclePaperImage p
          ↑j).cleanBranchSystemValue
      ↑i ↑j
    theorem QuantumBlockEncoding.GHL2025.functionOraclePaperMatrix_nonCleanInput_entry
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (i j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (hInputNonClean :
    (QuantumBlockEncoding.GHL2025.functionOraclePaperImage
          p ↑j).cleanWorkspaceBranch =
      false) :
  QuantumBlockEncoding.GHL2025.functionOraclePaperMatrix
      p i j =
    QuantumBlockEncoding.GHL2025.functionOracleOrthogonalEntry
      p
      (QuantumBlockEncoding.GHL2025.functionOraclePaperImage
          p ↑j).cleanBranchSystemValue
      ↑i ↑j
    Non-clean input columns are left in the symbolic `O_f` completion branch.
Definition10.2.291
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “function oracle matrix”. Helper-only O_f diagonal matrix: records function values f(x_j) on the diagonal.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Helper-only O_f diagonal matrix: records function values f(x_j) on the diagonal. For each compound basis state |j⟩, extracts the system register value i and sets the diagonal entry to 'robinFunctionValue n i' = Coeff.symbol "f_x_i". All off-diagonal entries are zero. The entry depends only on the system register (grid point index), not on the sparse index. The paper's O_f (Theorem amplitude-oracle for piece-wise polynomial function, main.tex:870-910) encodes f(x_j)/N_f via amplitude oracle. The 1/N_f normalization is absorbed into the block-encoding normalizer α = N_D · N_f · κ. This diagonal matrix is not the paper image; the paper-level clean branch and orthogonal-component obligations are recorded by 'functionOraclePaperImage', and the active gate keeps 'unitary.proved := false'. main.tex:870-910 -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:5365. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.291●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.functionOracleMatrix
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.Matrix
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
    QuantumBlockEncoding.Coeff
    def QuantumBlockEncoding.GHL2025.functionOracleMatrix
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.Matrix
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p))
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p))
    QuantumBlockEncoding.Coeff
    Helper-only O_f diagonal matrix: records function values f(x_j) on the diagonal.

For each compound basis state |j⟩, extracts the system register value i
and sets the diagonal entry to `robinFunctionValue n i` = Coeff.symbol "f_x_i".
All off-diagonal entries are zero.  The entry depends only on the system
register (grid point index), not on the sparse index.

The paper's O_f (Theorem amplitude-oracle for piece-wise polynomial function,
main.tex:870-910) encodes f(x_j)/N_f via amplitude oracle.  The 1/N_f
normalization is absorbed into the block-encoding normalizer
α = N_D · N_f · κ.  This diagonal matrix is not the paper image; the
paper-level clean branch and orthogonal-component obligations are recorded by
`functionOraclePaperImage`, and the active gate keeps `unitary.proved := false`.

main.tex:870-910 -
Definition10.2.292
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “one term robin gate o f”. Gate matrix for 'O_f' using the faithful paper-image matrix skeleton.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Gate matrix for 'O_f' using the faithful paper-image matrix skeleton. The active matrix now exposes the clean 'm_f' branch amplitude from 'functionOraclePaperImage'. The legacy diagonal helper 'functionOracleMatrix' remains available only as a function-value data check. Unitarity, amplitude correctness, the 'N_f' bound, and the orthogonal completion are still unproved. Guseynov-Huang-Liu 2025, Theorem amplitude-oracle for piece-wise polynomial function and Fig. 1-term Robin, arXiv:2506.20478.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:5384. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.292●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_f
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GateMatrix QuantumBlockEncoding.Coeff
    (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)
    def QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_f
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GateMatrix
    QuantumBlockEncoding.Coeff
    (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
      p)
    Gate matrix for `O_f` using the faithful paper-image matrix skeleton.

The active matrix now exposes the clean `m_f` branch amplitude from
`functionOraclePaperImage`.  The legacy diagonal helper `functionOracleMatrix`
remains available only as a function-value data check.  Unitarity, amplitude
correctness, the `N_f` bound, and the orthogonal completion are still unproved.
Guseynov-Huang-Liu 2025, Theorem amplitude-oracle for piece-wise polynomial
function and Fig. 1-term Robin, arXiv:2506.20478.
Definition10.2.293
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “swap oracle matrix”. Honest SWAP matrix: permutation matrix swapping the system register (n qubits at bits [1, 1+n)) with the O_D^BS register (n qubits at bits [1+n, 1+2n)).

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Honest SWAP matrix: permutation matrix swapping the system register (n qubits at bits [1, 1+n)) with the O_D^BS register (n qubits at bits [1+n, 1+2n)). For each basis state |j⟩: - Extract block1 = bits [1, 1+n) of j (system register value) - Extract block2 = bits [1+n, 1+2n) of j (O_D^BS register value) - diff = block1 XOR block2 - Swapped index = j XOR (diff <<< 1) XOR (diff <<< (1+n)) When block1 = block2 the SWAP is the identity. All bits outside the two n-qubit blocks (ancilla bit 0, indicator bit 1+2n, mf MSBs) are preserved. The SWAP image-level proof is now promoted: 'swapOracleImage' is proved self-inverse, 'swapOracleMatrix' is proved a finite permutation matrix, and 'oneTermRobinGate_SWAP.unitary.proved = true'. figure:1_term_ROBIN, main.tex:1140 -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:5410. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.293●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.swapOracleMatrix
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.Matrix
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
    QuantumBlockEncoding.Coeff
    def QuantumBlockEncoding.GHL2025.swapOracleMatrix
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.Matrix
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p))
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p))
    QuantumBlockEncoding.Coeff
    Honest SWAP matrix: permutation matrix swapping the system register
(n qubits at bits [1, 1+n)) with the O_D^BS register (n qubits at bits [1+n, 1+2n)).

For each basis state |j⟩:
  - Extract block1 = bits [1, 1+n) of j  (system register value)
  - Extract block2 = bits [1+n, 1+2n) of j (O_D^BS register value)
  - diff = block1 XOR block2
  - Swapped index = j XOR (diff <<< 1) XOR (diff <<< (1+n))

When block1 = block2 the SWAP is the identity.  All bits outside the two
n-qubit blocks (ancilla bit 0, indicator bit 1+2n, mf MSBs) are preserved.

  The SWAP image-level proof is now promoted: `swapOracleImage` is proved
  self-inverse, `swapOracleMatrix` is proved a finite permutation matrix, and
  `oneTermRobinGate_SWAP.unitary.proved = true`.
  figure:1_term_ROBIN, main.tex:1140 -
Definition10.2.294
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “swap oracle image”. Image function for the SWAP oracle: swaps two n-qubit register blocks.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Image function for the SWAP oracle: swaps two n-qubit register blocks. For each basis state j, swaps block1 (bits [1,1+n)) with block2 (bits [1+n,1+2n)) by XORing with the block difference shifted to each block position. main.tex:1140 -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:5426. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.294●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.swapOracleImage
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) : ℕ
    def QuantumBlockEncoding.GHL2025.swapOracleImage
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) : ℕ
    Image function for the SWAP oracle: swaps two n-qubit register blocks.
For each basis state j, swaps block1 (bits [1,1+n)) with block2 (bits [1+n,1+2n))
by XORing with the block difference shifted to each block position.
main.tex:1140 -
Definition10.2.295
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “swap oracle diff”. The n-bit XOR difference between the two register blocks exchanged by SWAP.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The n-bit XOR difference between the two register blocks exchanged by SWAP. This is the reusable proof-DAG interface for the SWAP image route: preservation of this value after one SWAP is the local ingredient for self-inverse and later finite-domain permutation proofs.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:5441. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.295●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.swapOracleDiff
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) : ℕ
    def QuantumBlockEncoding.GHL2025.swapOracleDiff
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) : ℕ
    The n-bit XOR difference between the two register blocks exchanged by SWAP.

This is the reusable proof-DAG interface for the SWAP image route: preservation
of this value after one SWAP is the local ingredient for self-inverse and later
finite-domain permutation proofs.
Theorem10.2.296
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “swap oracle image eq xor diff”; the hypotheses and conclusion in the code panel fix its exact scope. The SWAP image is the source index XORed by the same difference in both blocks.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The SWAP image is the source index XORed by the same difference in both blocks.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5449. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.296●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.swapOracleImage_eq_xor_diff
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  QuantumBlockEncoding.GHL2025.swapOracleImage p j =
    j ^^^ QuantumBlockEncoding.GHL2025.swapOracleDiff p j <<< 1 ^^^
      QuantumBlockEncoding.GHL2025.swapOracleDiff p j <<< (1 + p.n)
    theorem QuantumBlockEncoding.GHL2025.swapOracleImage_eq_xor_diff
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  QuantumBlockEncoding.GHL2025.swapOracleImage
      p j =
    j ^^^
        QuantumBlockEncoding.GHL2025.swapOracleDiff
            p j <<<
          1 ^^^
      QuantumBlockEncoding.GHL2025.swapOracleDiff
          p j <<<
        (1 + p.n)
    The SWAP image is the source index XORed by the same difference in both blocks.
Theorem10.2.297
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “swap oracle matrix eq image”; the hypotheses and conclusion in the code panel fix its exact scope. swapOracleMatrix entry equals image function check.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. swapOracleMatrix entry equals image function check.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5456. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.297●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.swapOracleMatrix_eq_image
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (i j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))) :
  QuantumBlockEncoding.GHL2025.swapOracleMatrix p i j =
    if ↑i = QuantumBlockEncoding.GHL2025.swapOracleImage p ↑j then
      QuantumBlockEncoding.Coeff.rat 1
    else QuantumBlockEncoding.Coeff.rat 0
    theorem QuantumBlockEncoding.GHL2025.swapOracleMatrix_eq_image
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (i j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p))) :
  QuantumBlockEncoding.GHL2025.swapOracleMatrix
      p i j =
    if
        ↑i =
          QuantumBlockEncoding.GHL2025.swapOracleImage
            p ↑j then
      QuantumBlockEncoding.Coeff.rat 1
    else QuantumBlockEncoding.Coeff.rat 0
    swapOracleMatrix entry equals image function check.
Definition10.2.298
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “one term robin gate swap”. Gate matrix for SWAP using the honest permutation matrix.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Gate matrix for SWAP using the honest permutation matrix. Swaps system register (bits [1,n+1)) with O_D^BS register (bits [n+1,2n+1)). Unitarity is backed by the proof-DAG permutation bridge below: 'swapOracleMatrix_is_permutation'. main.tex:1140 -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:5468. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.298●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.oneTermRobinGate_SWAP
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GateMatrix QuantumBlockEncoding.Coeff
    (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)
    def QuantumBlockEncoding.GHL2025.oneTermRobinGate_SWAP
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GateMatrix
    QuantumBlockEncoding.Coeff
    (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
      p)
    Gate matrix for SWAP using the honest permutation matrix.
Swaps system register (bits [1,n+1)) with O_D^BS register (bits [n+1,2n+1)).
Unitarity is backed by the proof-DAG permutation bridge below:
`swapOracleMatrix_is_permutation`.
main.tex:1140 -
Definition10.2.299
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access dagger matrix”. Transpose-style matrix for O_D^BS, sharing the forward sparse-access image map.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Transpose-style matrix for O_D^BS, sharing the forward sparse-access image map. For each i: compute image(i) using the forward mapping, then check if j = image(i). This is the matrix transpose of bandedSparseAccessMatrix. The inverse/unitarity proof is blocked until the forward boundary column-map contract is reconciled. main.tex:1148 -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:5483. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.299●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessDaggerMatrix
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.Matrix
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
    QuantumBlockEncoding.Coeff
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessDaggerMatrix
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.Matrix
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p))
    (QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p))
    QuantumBlockEncoding.Coeff
    Transpose-style matrix for O_D^BS, sharing the forward sparse-access image map.
For each i: compute image(i) using the forward mapping, then check if j = image(i).
This is the matrix transpose of bandedSparseAccessMatrix.  The inverse/unitarity
proof is blocked until the forward boundary column-map contract is reconciled.
main.tex:1148 -
Definition10.2.300
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “one term robin gate o d bs dagger”. Gate matrix for '(O_D^BS)^†' using the transpose-style paper-image matrix.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Gate matrix for '(O_D^BS)^†' using the transpose-style paper-image matrix. This is paired with 'bandedSparseAccessPaperMatrix'; it does not prove that the transpose is a true inverse on the relevant post-SWAP states. figure:1_term_ROBIN and Lemma 1, arXiv:2506.20478.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:5505. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.300●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GateMatrix QuantumBlockEncoding.Coeff
    (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)
    def QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GateMatrix
    QuantumBlockEncoding.Coeff
    (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
      p)
    Gate matrix for `(O_D^BS)^†` using the transpose-style paper-image matrix.

This is paired with `bandedSparseAccessPaperMatrix`; it does not prove that the
transpose is a true inverse on the relevant post-SWAP states.
figure:1_term_ROBIN and Lemma 1, arXiv:2506.20478.
Theorem10.2.301
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin gate o d bs dagger image fin eq one”; the hypotheses and conclusion in the code panel fix its exact scope. Active '(O_D^BS)^†' gate entry paired with the finite forward image.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Active '(O_D^BS)^†' gate entry paired with the finite forward image.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5515. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.301●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger_imageFin_eq_one
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p ↑j <
      1 <<< p.n) :
  (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger p).matrix
      j
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageFin p j
        haddr) =
    QuantumBlockEncoding.Coeff.rat 1
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger_imageFin_eq_one
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
        p ↑j <
      1 <<< p.n) :
  (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
          p).matrix
      j
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageFin
        p j haddr) =
    QuantumBlockEncoding.Coeff.rat 1
    Active `(O_D^BS)^†` gate entry paired with the finite forward image.
Theorem10.2.302
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin gate o d bs dagger post swap entry of preimage”; the hypotheses and conclusion in the code panel fix its exact scope. Post-SWAP dagger entry from an explicitly supplied paper-image preimage.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Post-SWAP dagger entry from an explicitly supplied paper-image preimage. The hypothesis 'hpre' is the whole inverse-on-range input for this lemma: it does not prove that such a 'pre' exists, that it is unique, or that the dagger cleans the padded sparse-index register. The post-SWAP relation is recorded by 'hpost' for the cleanup proof-DAG interface, but the matrix entry itself is just the active transpose-style paper-image entry.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5534. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.302●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger_postSwap_entry_of_preimage
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source post pre :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (hpost :
    ↑post =
      QuantumBlockEncoding.GHL2025.swapOracleImage p
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p
          ↑source))
  (hpre :
    ↑post =
      QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p
        ↑pre) :
  (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger p).matrix
      pre post =
    QuantumBlockEncoding.Coeff.rat 1
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger_postSwap_entry_of_preimage
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source post pre :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (hpost :
    ↑post =
      QuantumBlockEncoding.GHL2025.swapOracleImage
        p
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
          p ↑source))
  (hpre :
    ↑post =
      QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
        p ↑pre) :
  (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
          p).matrix
      pre post =
    QuantumBlockEncoding.Coeff.rat 1
    Post-SWAP dagger entry from an explicitly supplied paper-image preimage.

The hypothesis `hpre` is the whole inverse-on-range input for this lemma:
it does not prove that such a `pre` exists, that it is unique, or that the
dagger cleans the padded sparse-index register.  The post-SWAP relation is
recorded by `hpost` for the cleanup proof-DAG interface, but the matrix entry
itself is just the active transpose-style paper-image entry.
Definition10.2.303
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “banded sparse access post swap cleanup”. A proposition-valued field is a requirement until a constructor supplies it. Proof-carrying interface for a supplied post-SWAP cleanup preimage.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Proof-carrying interface for a supplied post-SWAP cleanup preimage. The fields intentionally include the hypotheses that are not yet derived: 'postSwap', 'preimage', 'preCleanInput', and 'preAddressBound'. The record only packages consequences of those inputs: the active dagger entry and executable register-cleanup checks for the chosen preimage. Existence, uniqueness, and the paper-level 'daggerCleanup' obligation remain open.

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:5555. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.303●1 definition
  • structure(11 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.BandedSparseAccessPostSwapCleanup
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source post pre :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))) :
  Prop
    structure QuantumBlockEncoding.GHL2025.BandedSparseAccessPostSwapCleanup
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source post pre :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p))) :
  Prop
    Proof-carrying interface for a supplied post-SWAP cleanup preimage.

The fields intentionally include the hypotheses that are not yet derived:
`postSwap`, `preimage`, `preCleanInput`, and `preAddressBound`.  The record only
packages consequences of those inputs: the active dagger entry and executable
register-cleanup checks for the chosen preimage.  Existence, uniqueness, and
the paper-level `daggerCleanup` obligation remain open.

    Fields

    postSwap : ↑post =
  QuantumBlockEncoding.GHL2025.swapOracleImage p (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p ↑source)
    preimage : ↑post = QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p ↑pre
    preCleanInput : QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput p ↑pre = true
    preAddressBound : QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p ↑pre < 1 <<< p.n
    daggerEntry : (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger p).matrix pre post = QuantumBlockEncoding.Coeff.rat 1
    preRowPreserved : (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract p ↑pre).rowPreserved = true
    preAddressWritten : (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract p ↑pre).addressWritten = true
    preAddressInRange : (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract p ↑pre).addressInRange = true
    preImageNoSpill : (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperColumnContract p ↑pre).imageNoSpill = true
    postRow_eq_preRow : (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p ↑post).rowValue =
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p ↑pre).rowValue
    postOd_eq_preAddress : (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p ↑post).odRegisterValue =
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p ↑pre
Definition10.2.304
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access post swap cleanup of preimage”. Build the post-SWAP cleanup witness from an explicitly supplied preimage.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Build the post-SWAP cleanup witness from an explicitly supplied preimage. This is the fixed inverse-on-range interface for the next cleanup proof: it does not construct the preimage and does not promote any semantic proof flag.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:5585. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.304●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPostSwapCleanup_of_preimage
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source post pre :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (hpost :
    ↑post =
      QuantumBlockEncoding.GHL2025.swapOracleImage p
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p
          ↑source))
  (hpre :
    ↑post =
      QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p ↑pre)
  (hclean :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput p
        ↑pre =
      true)
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p ↑pre <
      1 <<< p.n) :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessPostSwapCleanup p
    source post pre
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPostSwapCleanup_of_preimage
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source post pre :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (hpost :
    ↑post =
      QuantumBlockEncoding.GHL2025.swapOracleImage
        p
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
          p ↑source))
  (hpre :
    ↑post =
      QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
        p ↑pre)
  (hclean :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput
        p ↑pre =
      true)
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
        p ↑pre <
      1 <<< p.n) :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessPostSwapCleanup
    p source post pre
    Build the post-SWAP cleanup witness from an explicitly supplied preimage.

This is the fixed inverse-on-range interface for the next cleanup proof: it
does not construct the preimage and does not promote any semantic proof flag.
Theorem10.2.305
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin gate o d bs image fin entry safety”; the hypotheses and conclusion in the code panel fix its exact scope. Reusable image witness for the active Lemma 1 'O_D^BS' gate pair.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Reusable image witness for the active Lemma 1 'O_D^BS' gate pair. This packages the forward entry, transpose-style dagger entry, row roundtrip, written-address roundtrip, and no-spill Boolean under the explicit n-bit address hypothesis. It is not an injectivity, inverse uniqueness, cleanup, or unitarity proof.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5626. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.305●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_imageFin_entrySafety
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p ↑j <
      1 <<< p.n) :
  (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS p).matrix
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageFin p
          j haddr)
        j =
      QuantumBlockEncoding.Coeff.rat 1 ∧
    (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
              p).matrix
          j
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageFin
            p j haddr) =
        QuantumBlockEncoding.Coeff.rat 1 ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
              (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
                p ↑j)).rowValue =
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
              p ↑j).rowValue ∧
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
                (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
                  p ↑j)).odRegisterValue =
            QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
              p ↑j ∧
          QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageNoSpill
              p ↑j =
            true
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_imageFin_entrySafety
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
        p ↑j <
      1 <<< p.n) :
  (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
            p).matrix
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageFin
          p j haddr)
        j =
      QuantumBlockEncoding.Coeff.rat 1 ∧
    (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
              p).matrix
          j
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageFin
            p j haddr) =
        QuantumBlockEncoding.Coeff.rat 1 ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
              p
              (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
                p ↑j)).rowValue =
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
              p ↑j).rowValue ∧
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
                p
                (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
                  p ↑j)).odRegisterValue =
            QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
              p ↑j ∧
          QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageNoSpill
              p ↑j =
            true
    Reusable image witness for the active Lemma 1 `O_D^BS` gate pair.

This packages the forward entry, transpose-style dagger entry, row roundtrip,
written-address roundtrip, and no-spill Boolean under the explicit n-bit
address hypothesis.  It is not an injectivity, inverse uniqueness, cleanup, or
unitarity proof.
Theorem10.2.306
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin gate o d bs global slot source entry safety”; the hypotheses and conclusion in the code panel fix its exact scope. Global-source specialization of the active Lemma 1 'O_D^BS' entry witness.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Global-source specialization of the active Lemma 1 'O_D^BS' entry witness. For a finite column in the faithful source domain, the global-slot source predicate supplies the clean padded input and sparse-slot bound, while the '2 <= n' parameter-family hypothesis supplies the n-bit address bound. The result packages the finite image index and paired forward/dagger entries, but does not prove injectivity, cleanup, or unitarity.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5660. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.306●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_globalSlotSource_entrySafety
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (hn : 2 ≤ p.n)
  (hsource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑j =
      true) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput p ↑j =
      true ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
            ↑j).sparseIndexValue <
        p.kappa ∧
      ∃ image,
        ↑image =
            QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p
              ↑j ∧
          (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
                    p).matrix
                image j =
              QuantumBlockEncoding.Coeff.rat 1 ∧
            (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
                      p).matrix
                  j image =
                QuantumBlockEncoding.Coeff.rat 1 ∧
              (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
                      p
                      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
                        p ↑j)).rowValue =
                  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
                      p ↑j).rowValue ∧
                (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
                        p
                        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
                          p ↑j)).odRegisterValue =
                    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
                      p ↑j ∧
                  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageNoSpill
                        p ↑j =
                      true ∧
                    (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
                              p).unitary.proved =
                        false ∧
                      (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
                              p).unitary.proved =
                        false
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_globalSlotSource_entrySafety
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (hn : 2 ≤ p.n)
  (hsource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑j =
      true) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput
        p ↑j =
      true ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
            p ↑j).sparseIndexValue <
        p.kappa ∧
      ∃ image,
        ↑image =
            QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
              p ↑j ∧
          (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
                    p).matrix
                image j =
              QuantumBlockEncoding.Coeff.rat
                1 ∧
            (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
                      p).matrix
                  j image =
                QuantumBlockEncoding.Coeff.rat
                  1 ∧
              (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
                      p
                      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
                        p ↑j)).rowValue =
                  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
                      p ↑j).rowValue ∧
                (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
                        p
                        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
                          p
                          ↑j)).odRegisterValue =
                    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
                      p ↑j ∧
                  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImageNoSpill
                        p ↑j =
                      true ∧
                    (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
                              p).unitary.proved =
                        false ∧
                      (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
                              p).unitary.proved =
                        false
    Global-source specialization of the active Lemma 1 `O_D^BS` entry witness.

For a finite column in the faithful source domain, the global-slot source
predicate supplies the clean padded input and sparse-slot bound, while the
`2 <= n` parameter-family hypothesis supplies the n-bit address bound.  The
result packages the finite image index and paired forward/dagger entries, but
does not prove injectivity, cleanup, or unitarity.
Definition10.2.307
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “one term robin gate matrix placeholders”. List of all 7 gate matrix placeholders for the one-term Robin circuit, in the same order as 'oneTermRobinCircuit'.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. List of all 7 gate matrix placeholders for the one-term Robin circuit, in the same order as 'oneTermRobinCircuit'. figure:1_term_ROBIN -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:5706. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.307●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.oneTermRobinGateMatrixPlaceholders
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  List
    (QuantumBlockEncoding.GateMatrix QuantumBlockEncoding.Coeff
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
    def QuantumBlockEncoding.GHL2025.oneTermRobinGateMatrixPlaceholders
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  List
    (QuantumBlockEncoding.GateMatrix
      QuantumBlockEncoding.Coeff
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p))
    List of all 7 gate matrix placeholders for the one-term Robin circuit,
in the same order as `oneTermRobinCircuit`.
figure:1_term_ROBIN -
Theorem10.2.308
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin placeholders match”; the hypotheses and conclusion in the code panel fix its exact scope. The placeholder gate matrices match the circuit gate labels.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The placeholder gate matrices match the circuit gate labels. This is trivially true because the placeholders were constructed with matching gate constructors. figure:1_term_ROBIN -

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5722. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.308●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinPlaceholdersMatch
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.gateMatricesMatchCircuit
      QuantumBlockEncoding.GHL2025.oneTermRobinCircuit
      (QuantumBlockEncoding.GHL2025.oneTermRobinGateMatrixPlaceholders
        p) =
    true
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinPlaceholdersMatch
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.gateMatricesMatchCircuit
      QuantumBlockEncoding.GHL2025.oneTermRobinCircuit
      (QuantumBlockEncoding.GHL2025.oneTermRobinGateMatrixPlaceholders
        p) =
    true
    The placeholder gate matrices match the circuit gate labels.
This is trivially true because the placeholders were constructed
with matching gate constructors.
figure:1_term_ROBIN -
Theorem10.2.309
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin gate matrix placeholders gate list”; the hypotheses and conclusion in the code panel fix its exact scope. The active matrix placeholder list uses the same gate order as Fig.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The active matrix placeholder list uses the same gate order as Fig. 1-term Robin and 'oneTermRobinCircuit'. This is a structural guard only: it prevents a later proof packet from keeping similar-looking proof flags while changing the circuit order.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5738. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.309●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGateMatrixPlaceholders_gateList
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  List.map (fun gateMatrix => gateMatrix.gate)
      (QuantumBlockEncoding.GHL2025.oneTermRobinGateMatrixPlaceholders
        p) =
    QuantumBlockEncoding.GHL2025.oneTermRobinCircuit
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGateMatrixPlaceholders_gateList
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  List.map
      (fun gateMatrix => gateMatrix.gate)
      (QuantumBlockEncoding.GHL2025.oneTermRobinGateMatrixPlaceholders
        p) =
    QuantumBlockEncoding.GHL2025.oneTermRobinCircuit
    The active matrix placeholder list uses the same gate order as Fig. 1-term
Robin and `oneTermRobinCircuit`.

This is a structural guard only: it prevents a later proof packet from keeping
similar-looking proof flags while changing the circuit order.
Theorem10.2.310
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin gate matrix placeholders unitary flags”; the hypotheses and conclusion in the code panel fix its exact scope. The active seven-gate matrix list keeps only the locally certified indicator and SWAP gates marked as proved.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The active seven-gate matrix list keeps only the locally certified indicator and SWAP gates marked as proved. This is a Phase 1 guard for Fig. 1-term Robin. It records the current gate-level proof flags without promoting the paper-oracle obligations for 'O_DT^S', 'Ry_boundary', 'O_D^BS', 'O_f', or '(O_D^BS)^dagger'.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5757. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.310●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGateMatrixPlaceholders_unitaryFlags
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  List.map (fun gateMatrix => gateMatrix.unitary.proved)
      (QuantumBlockEncoding.GHL2025.oneTermRobinGateMatrixPlaceholders
        p) =
    [true, false, false, false, false, true, false]
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGateMatrixPlaceholders_unitaryFlags
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  List.map
      (fun gateMatrix =>
        gateMatrix.unitary.proved)
      (QuantumBlockEncoding.GHL2025.oneTermRobinGateMatrixPlaceholders
        p) =
    [true, false, false, false, false,
      true, false]
    The active seven-gate matrix list keeps only the locally certified indicator
and SWAP gates marked as proved.

This is a Phase 1 guard for Fig. 1-term Robin.  It records the current
gate-level proof flags without promoting the paper-oracle obligations for
`O_DT^S`, `Ry_boundary`, `O_D^BS`, `O_f`, or `(O_D^BS)^dagger`.
Definition10.2.311
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “indicator oracle image”. Indicator oracle image function: for each basis state j, computes the image by XORing the indicator bit at position indPos when the system register value is in the bulk window [K1, K2].

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Indicator oracle image function: for each basis state j, computes the image by XORing the indicator bit at position indPos when the system register value is in the bulk window [K1, K2]. This is a self-inverse permutation. main.tex:1088-1099 -

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:5773. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.311●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.indicatorOracleImage
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) : ℕ
    def QuantumBlockEncoding.GHL2025.indicatorOracleImage
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) : ℕ
    Indicator oracle image function: for each basis state j, computes the image
by XORing the indicator bit at position indPos when the system register value
is in the bulk window [K1, K2]. This is a self-inverse permutation.
main.tex:1088-1099 -
Theorem10.2.312
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “indicator oracle matrix eq image”; the hypotheses and conclusion in the code panel fix its exact scope. The indicator oracle matrix entry is 1 exactly when i = indicatorOracleImage j.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The indicator oracle matrix entry is 1 exactly when i = indicatorOracleImage j. main.tex:1088-1099 -

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5785. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.312●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleMatrix_eq_image
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (i j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))) :
  QuantumBlockEncoding.GHL2025.indicatorOracleMatrix p i j =
    if ↑i = QuantumBlockEncoding.GHL2025.indicatorOracleImage p ↑j then
      QuantumBlockEncoding.Coeff.rat 1
    else QuantumBlockEncoding.Coeff.rat 0
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleMatrix_eq_image
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (i j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p))) :
  QuantumBlockEncoding.GHL2025.indicatorOracleMatrix
      p i j =
    if
        ↑i =
          QuantumBlockEncoding.GHL2025.indicatorOracleImage
            p ↑j then
      QuantumBlockEncoding.Coeff.rat 1
    else QuantumBlockEncoding.Coeff.rat 0
    The indicator oracle matrix entry is 1 exactly when i = indicatorOracleImage j.
main.tex:1088-1099 -
Theorem10.2.313
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “indicator oracle image self inverse n 1”; the hypotheses and conclusion in the code panel fix its exact scope. Self-inverse property for n=1: applying indicatorOracleImage twice returns the original value for all j in Fin domain (128 elements).

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Self-inverse property for n=1: applying indicatorOracleImage twice returns the original value for all j in Fin domain (128 elements). Checked by native_decide over the finite Fin type. main.tex:1088-1099 -

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5796. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.313●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleImage_self_inverse_n1
  (j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          { n := 1, kappa := 1, functionPieces := 1,
            polynomialDegreeCost := 1 }))) :
  QuantumBlockEncoding.GHL2025.indicatorOracleImage
      { n := 1, kappa := 1, functionPieces := 1,
        polynomialDegreeCost := 1 }
      (QuantumBlockEncoding.GHL2025.indicatorOracleImage
        { n := 1, kappa := 1, functionPieces := 1,
          polynomialDegreeCost := 1 }
        ↑j) =
    ↑j
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleImage_self_inverse_n1
  (j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          { n := 1, kappa := 1,
            functionPieces := 1,
            polynomialDegreeCost :=
              1 }))) :
  QuantumBlockEncoding.GHL2025.indicatorOracleImage
      { n := 1, kappa := 1,
        functionPieces := 1,
        polynomialDegreeCost := 1 }
      (QuantumBlockEncoding.GHL2025.indicatorOracleImage
        { n := 1, kappa := 1,
          functionPieces := 1,
          polynomialDegreeCost := 1 }
        ↑j) =
    ↑j
    Self-inverse property for n=1: applying indicatorOracleImage twice returns the
original value for all j in Fin domain (128 elements).
Checked by native_decide over the finite Fin type.
main.tex:1088-1099 -
Theorem10.2.314
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “indicator oracle image self inverse n 3”; the hypotheses and conclusion in the code panel fix its exact scope. Self-inverse property for n=3: applying indicatorOracleImage twice returns the original value for all j in Fin domain (8192 elements).

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Self-inverse property for n=3: applying indicatorOracleImage twice returns the original value for all j in Fin domain (8192 elements). Checked by native_decide over the finite Fin type. main.tex:1088-1099 -

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5810. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.314●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleImage_self_inverse_n3
  (j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          { n := 3, kappa := 7, functionPieces := 1,
            polynomialDegreeCost := 1 }))) :
  QuantumBlockEncoding.GHL2025.indicatorOracleImage
      { n := 3, kappa := 7, functionPieces := 1,
        polynomialDegreeCost := 1 }
      (QuantumBlockEncoding.GHL2025.indicatorOracleImage
        { n := 3, kappa := 7, functionPieces := 1,
          polynomialDegreeCost := 1 }
        ↑j) =
    ↑j
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleImage_self_inverse_n3
  (j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          { n := 3, kappa := 7,
            functionPieces := 1,
            polynomialDegreeCost :=
              1 }))) :
  QuantumBlockEncoding.GHL2025.indicatorOracleImage
      { n := 3, kappa := 7,
        functionPieces := 1,
        polynomialDegreeCost := 1 }
      (QuantumBlockEncoding.GHL2025.indicatorOracleImage
        { n := 3, kappa := 7,
          functionPieces := 1,
          polynomialDegreeCost := 1 }
        ↑j) =
    ↑j
    Self-inverse property for n=3: applying indicatorOracleImage twice returns the
original value for all j in Fin domain (8192 elements).
Checked by native_decide over the finite Fin type.
main.tex:1088-1099 -
Theorem10.2.315
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “indicator oracle image injective n 1”; the hypotheses and conclusion in the code panel fix its exact scope. Injectivity for n=1: derived from self-inverse property.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Injectivity for n=1: derived from self-inverse property. main.tex:1088-1099 -

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5822. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.315●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleImage_injective_n1
  {j₁ j₂ :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          { n := 1, kappa := 1, functionPieces := 1,
            polynomialDegreeCost := 1 }))}
  (h :
    QuantumBlockEncoding.GHL2025.indicatorOracleImage
        { n := 1, kappa := 1, functionPieces := 1,
          polynomialDegreeCost := 1 }
        ↑j₁ =
      QuantumBlockEncoding.GHL2025.indicatorOracleImage
        { n := 1, kappa := 1, functionPieces := 1,
          polynomialDegreeCost := 1 }
        ↑j₂) :
  j₁ = j₂
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleImage_injective_n1
  {j₁ j₂ :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          { n := 1, kappa := 1,
            functionPieces := 1,
            polynomialDegreeCost := 1 }))}
  (h :
    QuantumBlockEncoding.GHL2025.indicatorOracleImage
        { n := 1, kappa := 1,
          functionPieces := 1,
          polynomialDegreeCost := 1 }
        ↑j₁ =
      QuantumBlockEncoding.GHL2025.indicatorOracleImage
        { n := 1, kappa := 1,
          functionPieces := 1,
          polynomialDegreeCost := 1 }
        ↑j₂) :
  j₁ = j₂
    Injectivity for n=1: derived from self-inverse property.
main.tex:1088-1099 -
Theorem10.2.316
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “indicator oracle image injective n 3”; the hypotheses and conclusion in the code panel fix its exact scope. Injectivity for n=3: derived from self-inverse property.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Injectivity for n=3: derived from self-inverse property. main.tex:1088-1099 -

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5838. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.316●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleImage_injective_n3
  {j₁ j₂ :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          { n := 3, kappa := 7, functionPieces := 1,
            polynomialDegreeCost := 1 }))}
  (h :
    QuantumBlockEncoding.GHL2025.indicatorOracleImage
        { n := 3, kappa := 7, functionPieces := 1,
          polynomialDegreeCost := 1 }
        ↑j₁ =
      QuantumBlockEncoding.GHL2025.indicatorOracleImage
        { n := 3, kappa := 7, functionPieces := 1,
          polynomialDegreeCost := 1 }
        ↑j₂) :
  j₁ = j₂
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleImage_injective_n3
  {j₁ j₂ :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          { n := 3, kappa := 7,
            functionPieces := 1,
            polynomialDegreeCost := 1 }))}
  (h :
    QuantumBlockEncoding.GHL2025.indicatorOracleImage
        { n := 3, kappa := 7,
          functionPieces := 1,
          polynomialDegreeCost := 1 }
        ↑j₁ =
      QuantumBlockEncoding.GHL2025.indicatorOracleImage
        { n := 3, kappa := 7,
          functionPieces := 1,
          polynomialDegreeCost := 1 }
        ↑j₂) :
  j₁ = j₂
    Injectivity for n=3: derived from self-inverse property.
main.tex:1088-1099 -
Theorem10.2.317
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “shift left land mask eq zero”; the hypotheses and conclusion in the code panel fix its exact scope. Cycle 12 helper: (b <<< pos) &&& ((1 <<< n) - 1) = 0 when pos >= n, because b <<< pos has all zeros in bits [0, pos) >= [0, n).

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Cycle 12 helper: (b <<< pos) &&& ((1 <<< n) - 1) = 0 when pos >= n, because b <<< pos has all zeros in bits [0, pos) >= [0, n).

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5855. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.317●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.shiftLeft_land_mask_eq_zero (b pos n : ℕ)
  (h : pos ≥ n) : b <<< pos &&& 1 <<< n - 1 = 0
    theorem QuantumBlockEncoding.GHL2025.shiftLeft_land_mask_eq_zero
  (b pos n : ℕ) (h : pos ≥ n) :
  b <<< pos &&& 1 <<< n - 1 = 0
    Cycle 12 helper: (b <<< pos) &&& ((1 <<< n) - 1) = 0 when pos >= n,
because b <<< pos has all zeros in bits [0, pos) >= [0, n).
Theorem10.2.318
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “xor shift preserve low”; the hypotheses and conclusion in the code panel fix its exact scope. Cycle 12 helper: XOR with a value shifted left by 'pos' preserves the low 'n' bits when 'pos >= n'.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Cycle 12 helper: XOR with a value shifted left by 'pos' preserves the low 'n' bits when 'pos >= n'. Uses AND-XOR distributivity and the zero mask lemma.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5872. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.318●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.xor_shift_preserve_low (x b pos n : ℕ)
  (h : pos ≥ n) : (x ^^^ b <<< pos) &&& 1 <<< n - 1 = x &&& 1 <<< n - 1
    theorem QuantumBlockEncoding.GHL2025.xor_shift_preserve_low
  (x b pos n : ℕ) (h : pos ≥ n) :
  (x ^^^ b <<< pos) &&& 1 <<< n - 1 =
    x &&& 1 <<< n - 1
    Cycle 12 helper: XOR with a value shifted left by `pos` preserves the low `n` bits
when `pos >= n`.  Uses AND-XOR distributivity and the zero mask lemma.
Theorem10.2.319
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “xor shift preserve shift low”; the hypotheses and conclusion in the code panel fix its exact scope. Cycle 12 helper: XOR with a high-shifted value preserves low bits after right-shifting.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Cycle 12 helper: XOR with a high-shifted value preserves low bits after right-shifting. ((x ^^^ (b <<< pos)) >>> 1) &&& ((1 <<< n) - 1) = (x >>> 1) &&& ((1 <<< n) - 1) when pos >= 1 + n.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5883. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.319●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.xor_shift_preserve_shift_low
  (x b pos n : ℕ) (h : pos ≥ 1 + n) :
  (x ^^^ b <<< pos) >>> 1 &&& 1 <<< n - 1 = x >>> 1 &&& 1 <<< n - 1
    theorem QuantumBlockEncoding.GHL2025.xor_shift_preserve_shift_low
  (x b pos n : ℕ) (h : pos ≥ 1 + n) :
  (x ^^^ b <<< pos) >>> 1 &&&
      1 <<< n - 1 =
    x >>> 1 &&& 1 <<< n - 1
    Cycle 12 helper: XOR with a high-shifted value preserves low bits after right-shifting.
((x ^^^ (b <<< pos)) >>> 1) &&& ((1 <<< n) - 1) = (x >>> 1) &&& ((1 <<< n) - 1)
when pos >= 1 + n.
Theorem10.2.320
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “swap oracle diff lt two pow”; the hypotheses and conclusion in the code panel fix its exact scope. SWAP proof-DAG helper: the XOR difference between the two n-bit blocks is itself an n-bit value.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. SWAP proof-DAG helper: the XOR difference between the two n-bit blocks is itself an n-bit value. main.tex:1140 -

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5908. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.320●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.swapOracleDiff_lt_two_pow
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  have n := p.n;
  have blockMask := 1 <<< n - 1;
  have block1 := j >>> 1 &&& blockMask;
  have block2 := j >>> (1 + n) &&& blockMask;
  block1 ^^^ block2 < 2 ^ n
    theorem QuantumBlockEncoding.GHL2025.swapOracleDiff_lt_two_pow
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  have n := p.n;
  have blockMask := 1 <<< n - 1;
  have block1 := j >>> 1 &&& blockMask;
  have block2 :=
    j >>> (1 + n) &&& blockMask;
  block1 ^^^ block2 < 2 ^ n
    SWAP proof-DAG helper: the XOR difference between the two n-bit blocks is
itself an n-bit value.
main.tex:1140 -
Theorem10.2.321
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “swap oracle diff shift right eq zero”; the hypotheses and conclusion in the code panel fix its exact scope. SWAP proof-DAG helper: right-shifting the n-bit block difference by n removes it.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. SWAP proof-DAG helper: right-shifting the n-bit block difference by n removes it. main.tex:1140 -

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5926. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.321●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.swapOracleDiff_shiftRight_eq_zero
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  have n := p.n;
  have blockMask := 1 <<< n - 1;
  have block1 := j >>> 1 &&& blockMask;
  have block2 := j >>> (1 + n) &&& blockMask;
  have diff := block1 ^^^ block2;
  diff >>> n = 0
    theorem QuantumBlockEncoding.GHL2025.swapOracleDiff_shiftRight_eq_zero
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  have n := p.n;
  have blockMask := 1 <<< n - 1;
  have block1 := j >>> 1 &&& blockMask;
  have block2 :=
    j >>> (1 + n) &&& blockMask;
  have diff := block1 ^^^ block2;
  diff >>> n = 0
    SWAP proof-DAG helper: right-shifting the n-bit block difference by n removes it.
main.tex:1140 -
Theorem10.2.322
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “swap oracle diff shift left mask eq zero”; the hypotheses and conclusion in the code panel fix its exact scope. SWAP proof-DAG helper: shifting the block difference into the high block leaves zero in the low n-bit mask.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. SWAP proof-DAG helper: shifting the block difference into the high block leaves zero in the low n-bit mask. main.tex:1140 -

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5941. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.322●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.swapOracleDiff_shiftLeft_mask_eq_zero
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  have n := p.n;
  have blockMask := 1 <<< n - 1;
  have block1 := j >>> 1 &&& blockMask;
  have block2 := j >>> (1 + n) &&& blockMask;
  have diff := block1 ^^^ block2;
  diff <<< n &&& blockMask = 0
    theorem QuantumBlockEncoding.GHL2025.swapOracleDiff_shiftLeft_mask_eq_zero
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  have n := p.n;
  have blockMask := 1 <<< n - 1;
  have block1 := j >>> 1 &&& blockMask;
  have block2 :=
    j >>> (1 + n) &&& blockMask;
  have diff := block1 ^^^ block2;
  diff <<< n &&& blockMask = 0
    SWAP proof-DAG helper: shifting the block difference into the high block leaves
zero in the low n-bit mask.
main.tex:1140 -
Theorem10.2.323
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “shift left lt two pow of lt”; the hypotheses and conclusion in the code panel fix its exact scope. Shifting a bounded value into a register block keeps it inside the total basis width.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Shifting a bounded value into a register block keeps it inside the total basis width.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5954. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.323●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.shiftLeft_lt_two_pow_of_lt
  {x width shift total : ℕ} (hx : x < 2 ^ width)
  (hblock : width + shift ≤ total) : x <<< shift < 2 ^ total
    theorem QuantumBlockEncoding.GHL2025.shiftLeft_lt_two_pow_of_lt
  {x width shift total : ℕ}
  (hx : x < 2 ^ width)
  (hblock : width + shift ≤ total) :
  x <<< shift < 2 ^ total
    Shifting a bounded value into a register block keeps it inside the total basis width.
Theorem10.2.324
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “swap oracle image lt qubit dim”; the hypotheses and conclusion in the code panel fix its exact scope. SWAP proof-DAG range block: the image of the register-block SWAP stays inside the same full finite basis.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. SWAP proof-DAG range block: the image of the register-block SWAP stays inside the same full finite basis. This is only a range lemma; the finite permutation bridge is proved separately by 'swapOracleMatrix_is_permutation'.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:5972. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.324●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.swapOracleImage_lt_qubitDim
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) {j : ℕ}
  (hj :
    j <
      QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)) :
  QuantumBlockEncoding.GHL2025.swapOracleImage p j <
    QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)
    theorem QuantumBlockEncoding.GHL2025.swapOracleImage_lt_qubitDim
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  {j : ℕ}
  (hj :
    j <
      QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)) :
  QuantumBlockEncoding.GHL2025.swapOracleImage
      p j <
    QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p)
    SWAP proof-DAG range block: the image of the register-block SWAP stays inside
the same full finite basis.  This is only a range lemma; the finite
permutation bridge is proved separately by `swapOracleMatrix_is_permutation`.
Theorem10.2.325
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “swap oracle image block 1 eq block 2”; the hypotheses and conclusion in the code panel fix its exact scope. SWAP proof-DAG block: after 'swapOracleImage', the low n-bit register equals the old high n-bit register.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. SWAP proof-DAG block: after 'swapOracleImage', the low n-bit register equals the old high n-bit register. This is the first register-level bit-slice lemma needed for the eventual SWAP self-inverse/permutation proof. main.tex:1140 -

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6003. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.325●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.swapOracleImage_block1_eq_block2
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  have n := p.n;
  have blockMask := 1 <<< n - 1;
  QuantumBlockEncoding.GHL2025.swapOracleImage p j >>> 1 &&& blockMask =
    j >>> (1 + n) &&& blockMask
    theorem QuantumBlockEncoding.GHL2025.swapOracleImage_block1_eq_block2
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  have n := p.n;
  have blockMask := 1 <<< n - 1;
  QuantumBlockEncoding.GHL2025.swapOracleImage
          p j >>>
        1 &&&
      blockMask =
    j >>> (1 + n) &&& blockMask
    SWAP proof-DAG block: after `swapOracleImage`, the low n-bit register equals
the old high n-bit register.  This is the first register-level bit-slice lemma
needed for the eventual SWAP self-inverse/permutation proof.
main.tex:1140 -
Theorem10.2.326
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “swap oracle image block 2 eq block 1”; the hypotheses and conclusion in the code panel fix its exact scope. SWAP proof-DAG block: after 'swapOracleImage', the high n-bit register equals the old low n-bit register.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. SWAP proof-DAG block: after 'swapOracleImage', the high n-bit register equals the old low n-bit register. This is the symmetric register equation paired with 'swapOracleImage_block1_eq_block2'; it is still only a bit-slice block, not a SWAP unitarity proof. main.tex:1140 -

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6031. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.326●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.swapOracleImage_block2_eq_block1
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  have n := p.n;
  have blockMask := 1 <<< n - 1;
  QuantumBlockEncoding.GHL2025.swapOracleImage p j >>> (1 + n) &&&
      blockMask =
    j >>> 1 &&& blockMask
    theorem QuantumBlockEncoding.GHL2025.swapOracleImage_block2_eq_block1
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  have n := p.n;
  have blockMask := 1 <<< n - 1;
  QuantumBlockEncoding.GHL2025.swapOracleImage
          p j >>>
        (1 + n) &&&
      blockMask =
    j >>> 1 &&& blockMask
    SWAP proof-DAG block: after `swapOracleImage`, the high n-bit register equals
the old low n-bit register.  This is the symmetric register equation paired
with `swapOracleImage_block1_eq_block2`; it is still only a bit-slice block,
not a SWAP unitarity proof.
main.tex:1140 -
Theorem10.2.327
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “swap oracle diff preserved”; the hypotheses and conclusion in the code panel fix its exact scope. SWAP proof-DAG block: the XOR difference between the two exchanged registers is preserved by one SWAP application.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. SWAP proof-DAG block: the XOR difference between the two exchanged registers is preserved by one SWAP application. This uses only the two register block equations and is independent of any gate unitarity flag.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6065. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.327●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.swapOracleDiff_preserved
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  QuantumBlockEncoding.GHL2025.swapOracleDiff p
      (QuantumBlockEncoding.GHL2025.swapOracleImage p j) =
    QuantumBlockEncoding.GHL2025.swapOracleDiff p j
    theorem QuantumBlockEncoding.GHL2025.swapOracleDiff_preserved
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  QuantumBlockEncoding.GHL2025.swapOracleDiff
      p
      (QuantumBlockEncoding.GHL2025.swapOracleImage
        p j) =
    QuantumBlockEncoding.GHL2025.swapOracleDiff
      p j
    SWAP proof-DAG block: the XOR difference between the two exchanged registers is
preserved by one SWAP application.  This uses only the two register block
equations and is independent of any gate unitarity flag.
Theorem10.2.328
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “xor two shifted masks cancel”; the hypotheses and conclusion in the code panel fix its exact scope. XORing the same two shifted masks twice cancels them bitwise.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. XORing the same two shifted masks twice cancels them bitwise.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6074. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.328●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.xor_two_shifted_masks_cancel
  (j diff n : ℕ) :
  j ^^^ diff <<< 1 ^^^ diff <<< (1 + n) ^^^ diff <<< 1 ^^^
      diff <<< (1 + n) =
    j
    theorem QuantumBlockEncoding.GHL2025.xor_two_shifted_masks_cancel
  (j diff n : ℕ) :
  j ^^^ diff <<< 1 ^^^
          diff <<< (1 + n) ^^^
        diff <<< 1 ^^^
      diff <<< (1 + n) =
    j
    XORing the same two shifted masks twice cancels them bitwise.
Theorem10.2.329
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “swap oracle image self inverse”; the hypotheses and conclusion in the code panel fix its exact scope. SWAP proof-DAG block: the image function is self-inverse.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. SWAP proof-DAG block: the image function is self-inverse. This is the arithmetic image fact reused by the finite permutation-matrix bridge.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6091. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.329●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.swapOracleImage_self_inverse
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  QuantumBlockEncoding.GHL2025.swapOracleImage p
      (QuantumBlockEncoding.GHL2025.swapOracleImage p j) =
    j
    theorem QuantumBlockEncoding.GHL2025.swapOracleImage_self_inverse
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  QuantumBlockEncoding.GHL2025.swapOracleImage
      p
      (QuantumBlockEncoding.GHL2025.swapOracleImage
        p j) =
    j
    SWAP proof-DAG block: the image function is self-inverse.

This is the arithmetic image fact reused by the finite permutation-matrix
bridge.
Theorem10.2.330
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “swap oracle image injective”; the hypotheses and conclusion in the code panel fix its exact scope. SWAP proof-DAG block: injectivity of the image function, derived from the self-inverse arithmetic block without opening the bit-slice proof again.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. SWAP proof-DAG block: injectivity of the image function, derived from the self-inverse arithmetic block without opening the bit-slice proof again.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6102. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.330●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.swapOracleImage_injective
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) {j₁ j₂ : ℕ}
  (h :
    QuantumBlockEncoding.GHL2025.swapOracleImage p j₁ =
      QuantumBlockEncoding.GHL2025.swapOracleImage p j₂) :
  j₁ = j₂
    theorem QuantumBlockEncoding.GHL2025.swapOracleImage_injective
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  {j₁ j₂ : ℕ}
  (h :
    QuantumBlockEncoding.GHL2025.swapOracleImage
        p j₁ =
      QuantumBlockEncoding.GHL2025.swapOracleImage
        p j₂) :
  j₁ = j₂
    SWAP proof-DAG block: injectivity of the image function, derived from the
self-inverse arithmetic block without opening the bit-slice proof again.
Theorem10.2.331
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “swap oracle image bijective”; the hypotheses and conclusion in the code panel fix its exact scope. SWAP proof-DAG block: bijectivity of 'swapOracleImage' on the finite full Hilbert-space basis.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. SWAP proof-DAG block: bijectivity of 'swapOracleImage' on the finite full Hilbert-space basis. The finite map uses 'swapOracleImage_lt_qubitDim' for the 'Fin' constructor and 'swapOracleImage_self_inverse' for both injectivity and surjectivity.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6116. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.331●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.swapOracleImage_bijective
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  (∀
      (a b :
        Fin
          (QuantumBlockEncoding.qubitDim
            (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))),
      ⟨QuantumBlockEncoding.GHL2025.swapOracleImage p ↑a, ⋯⟩ =
          ⟨QuantumBlockEncoding.GHL2025.swapOracleImage p ↑b, ⋯⟩ →
        a = b) ∧
    ∀
      (y :
        Fin
          (QuantumBlockEncoding.qubitDim
            (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))),
      ∃ x, ⟨QuantumBlockEncoding.GHL2025.swapOracleImage p ↑x, ⋯⟩ = y
    theorem QuantumBlockEncoding.GHL2025.swapOracleImage_bijective
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  (∀
      (a b :
        Fin
          (QuantumBlockEncoding.qubitDim
            (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
              p))),
      ⟨QuantumBlockEncoding.GHL2025.swapOracleImage
              p ↑a,
            ⋯⟩ =
          ⟨QuantumBlockEncoding.GHL2025.swapOracleImage
              p ↑b,
            ⋯⟩ →
        a = b) ∧
    ∀
      (y :
        Fin
          (QuantumBlockEncoding.qubitDim
            (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
              p))),
      ∃ x,
        ⟨QuantumBlockEncoding.GHL2025.swapOracleImage
              p ↑x,
            ⋯⟩ =
          y
    SWAP proof-DAG block: bijectivity of `swapOracleImage` on the finite full
Hilbert-space basis.

The finite map uses `swapOracleImage_lt_qubitDim` for the `Fin` constructor and
`swapOracleImage_self_inverse` for both injectivity and surjectivity.
Theorem10.2.332
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “swap oracle matrix col has one”; the hypotheses and conclusion in the code panel fix its exact scope. For each SWAP matrix column, the row indexed by 'swapOracleImage' contains the unique '1' entry.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. For each SWAP matrix column, the row indexed by 'swapOracleImage' contains the unique '1' entry.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6139. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.332●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.swapOracleMatrix_col_has_one
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))) :
  QuantumBlockEncoding.GHL2025.swapOracleMatrix p
      ⟨QuantumBlockEncoding.GHL2025.swapOracleImage p ↑j, ⋯⟩ j =
    QuantumBlockEncoding.Coeff.rat 1
    theorem QuantumBlockEncoding.GHL2025.swapOracleMatrix_col_has_one
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p))) :
  QuantumBlockEncoding.GHL2025.swapOracleMatrix
      p
      ⟨QuantumBlockEncoding.GHL2025.swapOracleImage
          p ↑j,
        ⋯⟩
      j =
    QuantumBlockEncoding.Coeff.rat 1
    For each SWAP matrix column, the row indexed by `swapOracleImage` contains the
unique `1` entry.
Theorem10.2.333
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “swap oracle matrix col unique”; the hypotheses and conclusion in the code panel fix its exact scope. For each SWAP matrix column, any '1' entry must occur at the row indexed by 'swapOracleImage'.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. For each SWAP matrix column, any '1' entry must occur at the row indexed by 'swapOracleImage'.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6151. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.333●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.swapOracleMatrix_col_unique
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (i j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (h :
    QuantumBlockEncoding.GHL2025.swapOracleMatrix p i j =
      QuantumBlockEncoding.Coeff.rat 1) :
  i = ⟨QuantumBlockEncoding.GHL2025.swapOracleImage p ↑j, ⋯⟩
    theorem QuantumBlockEncoding.GHL2025.swapOracleMatrix_col_unique
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (i j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (h :
    QuantumBlockEncoding.GHL2025.swapOracleMatrix
        p i j =
      QuantumBlockEncoding.Coeff.rat 1) :
  i =
    ⟨QuantumBlockEncoding.GHL2025.swapOracleImage
        p ↑j,
      ⋯⟩
    For each SWAP matrix column, any `1` entry must occur at the row indexed by
`swapOracleImage`.
Theorem10.2.334
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “swap oracle matrix row has one”; the hypotheses and conclusion in the code panel fix its exact scope. Every SWAP matrix row has a '1' entry, by finite surjectivity.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Every SWAP matrix row has a '1' entry, by finite surjectivity.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6161. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.334●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.swapOracleMatrix_row_has_one
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (i :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))) :
  ∃ j,
    QuantumBlockEncoding.GHL2025.swapOracleMatrix p i j =
      QuantumBlockEncoding.Coeff.rat 1
    theorem QuantumBlockEncoding.GHL2025.swapOracleMatrix_row_has_one
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (i :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p))) :
  ∃ j,
    QuantumBlockEncoding.GHL2025.swapOracleMatrix
        p i j =
      QuantumBlockEncoding.Coeff.rat 1
    Every SWAP matrix row has a `1` entry, by finite surjectivity.
Theorem10.2.335
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “swap oracle matrix row unique”; the hypotheses and conclusion in the code panel fix its exact scope. Every SWAP matrix row has a unique '1' entry, by finite injectivity.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Every SWAP matrix row has a unique '1' entry, by finite injectivity.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6176. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.335●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.swapOracleMatrix_row_unique
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (i j₁ j₂ :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (h₁ :
    QuantumBlockEncoding.GHL2025.swapOracleMatrix p i j₁ =
      QuantumBlockEncoding.Coeff.rat 1)
  (h₂ :
    QuantumBlockEncoding.GHL2025.swapOracleMatrix p i j₂ =
      QuantumBlockEncoding.Coeff.rat 1) :
  j₁ = j₂
    theorem QuantumBlockEncoding.GHL2025.swapOracleMatrix_row_unique
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (i j₁ j₂ :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (h₁ :
    QuantumBlockEncoding.GHL2025.swapOracleMatrix
        p i j₁ =
      QuantumBlockEncoding.Coeff.rat 1)
  (h₂ :
    QuantumBlockEncoding.GHL2025.swapOracleMatrix
        p i j₂ =
      QuantumBlockEncoding.Coeff.rat 1) :
  j₁ = j₂
    Every SWAP matrix row has a unique `1` entry, by finite injectivity.
Theorem10.2.336
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “swap oracle matrix is permutation”; the hypotheses and conclusion in the code panel fix its exact scope. SWAP matrix is a finite permutation matrix: every row and column has exactly one entry equal to '1'.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. SWAP matrix is a finite permutation matrix: every row and column has exactly one entry equal to '1'. This closes the SWAP gate-level matrix-semantics bridge while leaving the paper-specific O_D^BS, O_f, LCU, and block-extraction obligations unchanged.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6194. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.336●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.swapOracleMatrix_is_permutation
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  (∀
      (i :
        Fin
          (QuantumBlockEncoding.qubitDim
            (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))),
      ∃ j,
        QuantumBlockEncoding.GHL2025.swapOracleMatrix p i j =
            QuantumBlockEncoding.Coeff.rat 1 ∧
          ∀
            (j' :
              Fin
                (QuantumBlockEncoding.qubitDim
                  (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
                    p))),
            QuantumBlockEncoding.GHL2025.swapOracleMatrix p i j' =
                QuantumBlockEncoding.Coeff.rat 1 →
              j' = j) ∧
    ∀
      (j :
        Fin
          (QuantumBlockEncoding.qubitDim
            (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))),
      ∃ i,
        QuantumBlockEncoding.GHL2025.swapOracleMatrix p i j =
            QuantumBlockEncoding.Coeff.rat 1 ∧
          ∀
            (i' :
              Fin
                (QuantumBlockEncoding.qubitDim
                  (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
                    p))),
            QuantumBlockEncoding.GHL2025.swapOracleMatrix p i' j =
                QuantumBlockEncoding.Coeff.rat 1 →
              i' = i
    theorem QuantumBlockEncoding.GHL2025.swapOracleMatrix_is_permutation
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  (∀
      (i :
        Fin
          (QuantumBlockEncoding.qubitDim
            (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
              p))),
      ∃ j,
        QuantumBlockEncoding.GHL2025.swapOracleMatrix
              p i j =
            QuantumBlockEncoding.Coeff.rat
              1 ∧
          ∀
            (j' :
              Fin
                (QuantumBlockEncoding.qubitDim
                  (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
                    p))),
            QuantumBlockEncoding.GHL2025.swapOracleMatrix
                  p i j' =
                QuantumBlockEncoding.Coeff.rat
                  1 →
              j' = j) ∧
    ∀
      (j :
        Fin
          (QuantumBlockEncoding.qubitDim
            (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
              p))),
      ∃ i,
        QuantumBlockEncoding.GHL2025.swapOracleMatrix
              p i j =
            QuantumBlockEncoding.Coeff.rat
              1 ∧
          ∀
            (i' :
              Fin
                (QuantumBlockEncoding.qubitDim
                  (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
                    p))),
            QuantumBlockEncoding.GHL2025.swapOracleMatrix
                  p i' j =
                QuantumBlockEncoding.Coeff.rat
                  1 →
              i' = i
    SWAP matrix is a finite permutation matrix: every row and column has exactly
one entry equal to `1`.

This closes the SWAP gate-level matrix-semantics bridge while leaving the
paper-specific O_D^BS, O_f, LCU, and block-extraction obligations unchanged.
Theorem10.2.337
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper post swap row value eq address”; the hypotheses and conclusion in the code panel fix its exact scope. After the active Lemma 1 paper image and the SWAP gate, the system-row register contains the paper address 'r_si'.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. After the active Lemma 1 paper image and the SWAP gate, the system-row register contains the paper address 'r_si'. This is a post-SWAP register equation under the same n-bit address hypothesis used by the finite image bridge; it does not construct a dagger preimage or promote cleanup.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6221. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.337●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwap_rowValue_eq_address
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ)
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p j <
      1 <<< p.n) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
        (QuantumBlockEncoding.GHL2025.swapOracleImage p
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p
            j))).rowValue =
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p j
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwap_rowValue_eq_address
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ)
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
        p j <
      1 <<< p.n) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
        p
        (QuantumBlockEncoding.GHL2025.swapOracleImage
          p
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
            p j))).rowValue =
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
      p j
    After the active Lemma 1 paper image and the SWAP gate, the system-row register
contains the paper address `r_si`.  This is a post-SWAP register equation under
the same n-bit address hypothesis used by the finite image bridge; it does not
construct a dagger preimage or promote cleanup.
Theorem10.2.338
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper post swap od register value eq row value”; the hypotheses and conclusion in the code panel fix its exact scope. After the active Lemma 1 paper image and the SWAP gate, the O_D register contains the original row value.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. After the active Lemma 1 paper image and the SWAP gate, the O_D register contains the original row value. This is the second post-SWAP register equation needed before inverse-on-range cleanup search.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6238. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.338●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwap_odRegisterValue_eq_rowValue
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
        (QuantumBlockEncoding.GHL2025.swapOracleImage p
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p
            j))).odRegisterValue =
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
        j).rowValue
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwap_odRegisterValue_eq_rowValue
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
        p
        (QuantumBlockEncoding.GHL2025.swapOracleImage
          p
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
            p j))).odRegisterValue =
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
        p j).rowValue
    After the active Lemma 1 paper image and the SWAP gate, the O_D register
contains the original row value.  This is the second post-SWAP register
equation needed before inverse-on-range cleanup search.
Theorem10.2.339
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper post swap image lt qubit dim of address lt”; the hypotheses and conclusion in the code panel fix its exact scope. After the active paper image and SWAP, the post-SWAP column is still a finite basis index whenever the source column is finite and the written paper address is n-bit.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. After the active paper image and SWAP, the post-SWAP column is still a finite basis index whenever the source column is finite and the written paper address is n-bit. This does not prove inverse-on-range or cleanup.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6254. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.339●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapImage_lt_qubitDim_of_address_lt
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p
        ↑source <
      1 <<< p.n) :
  QuantumBlockEncoding.GHL2025.swapOracleImage p
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p
        ↑source) <
    QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapImage_lt_qubitDim_of_address_lt
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (haddr :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
        p ↑source <
      1 <<< p.n) :
  QuantumBlockEncoding.GHL2025.swapOracleImage
      p
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
        p ↑source) <
    QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p)
    After the active paper image and SWAP, the post-SWAP column is still a finite
basis index whenever the source column is finite and the written paper address
is n-bit.  This does not prove inverse-on-range or cleanup.
Definition10.2.340
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access paper splice od register”. Replace the 'O_D^BS' n-bit register of a compound index while preserving the low ancilla/system block and all high-tail bits.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Replace the 'O_D^BS' n-bit register of a compound index while preserving the low ancilla/system block and all high-tail bits. This is the local splice used to build a post-SWAP cleanup preimage candidate. It does not assert that the chosen 'odValue' is the correct reverse address.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:6274. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.340●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j odValue : ℕ) : ℕ
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j odValue : ℕ) : ℕ
    Replace the `O_D^BS` n-bit register of a compound index while preserving the
low ancilla/system block and all high-tail bits.

This is the local splice used to build a post-SWAP cleanup preimage candidate.
It does not assert that the chosen `odValue` is the correct reverse address.
Theorem10.2.341
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper image eq splice”; the hypotheses and conclusion in the code panel fix its exact scope. The paper image is the O_D-register splice with the computed paper address.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The paper image is the O_D-register splice with the computed paper address.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6283. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.341●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage_eq_splice
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p j =
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister
      p j
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress p j)
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage_eq_splice
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
      p j =
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister
      p j
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperAddress
        p j)
    The paper image is the O_D-register splice with the computed paper address.
Theorem10.2.342
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper splice od register low block lt high base of od value lt”; the hypotheses and conclusion in the code panel fix its exact scope. The spliced low-and-O_D block fits below the high-tail boundary for n-bit O_D values.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The spliced low-and-O_D block fits below the high-tail boundary for n-bit O_D values.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6291. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.342●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister_lowBlock_lt_highBase_of_odValue_lt
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j odValue : ℕ) (hod : odValue < 1 <<< p.n) :
  have lowWidth := 1 + p.n;
  have highWidth := 1 + 2 * p.n;
  have lowBase := 2 ^ lowWidth;
  have highBase := 2 ^ highWidth;
  have lowPrefix := j % lowBase;
  lowPrefix + odValue * lowBase < highBase
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister_lowBlock_lt_highBase_of_odValue_lt
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j odValue : ℕ)
  (hod : odValue < 1 <<< p.n) :
  have lowWidth := 1 + p.n;
  have highWidth := 1 + 2 * p.n;
  have lowBase := 2 ^ lowWidth;
  have highBase := 2 ^ highWidth;
  have lowPrefix := j % lowBase;
  lowPrefix + odValue * lowBase < highBase
    The spliced low-and-O_D block fits below the high-tail boundary for n-bit O_D values.
Theorem10.2.343
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper splice od register mod low base”; the hypotheses and conclusion in the code panel fix its exact scope. Splicing an O_D value preserves the low ancilla-and-row block.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Splicing an O_D value preserves the low ancilla-and-row block.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6329. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.343●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister_mod_lowBase
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j odValue : ℕ) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister p
        j odValue %
      2 ^ (1 + p.n) =
    j % 2 ^ (1 + p.n)
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister_mod_lowBase
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j odValue : ℕ) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister
        p j odValue %
      2 ^ (1 + p.n) =
    j % 2 ^ (1 + p.n)
    Splicing an O_D value preserves the low ancilla-and-row block.
Theorem10.2.344
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper splice od register div low base mod eq”; the hypotheses and conclusion in the code panel fix its exact scope. Splicing an n-bit O_D value exposes that value when the O_D register is extracted.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Splicing an n-bit O_D value exposes that value when the O_D register is extracted.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6364. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.344●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister_div_lowBase_mod_eq
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j odValue : ℕ) (hod : odValue < 1 <<< p.n) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister p
          j odValue /
        2 ^ (1 + p.n) %
      2 ^ p.n =
    odValue
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister_div_lowBase_mod_eq
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j odValue : ℕ)
  (hod : odValue < 1 <<< p.n) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister
          p j odValue /
        2 ^ (1 + p.n) %
      2 ^ p.n =
    odValue
    Splicing an n-bit O_D value exposes that value when the O_D register is extracted.
Theorem10.2.345
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper splice od register row value eq”; the hypotheses and conclusion in the code panel fix its exact scope. Splicing preserves the row field.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Splicing preserves the row field.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6407. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.345●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister_rowValue_eq
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j odValue : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister
          p j odValue)).rowValue =
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
        j).rowValue
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister_rowValue_eq
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j odValue : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
        p
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister
          p j odValue)).rowValue =
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
        p j).rowValue
    Splicing preserves the row field.
Theorem10.2.346
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper splice od register od register value eq”; the hypotheses and conclusion in the code panel fix its exact scope. Splicing an n-bit value into the O_D block makes that value the extracted O_D register.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Splicing an n-bit value into the O_D block makes that value the extracted O_D register.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6427. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.346●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister_odRegisterValue_eq
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j odValue : ℕ) (hod : odValue < 1 <<< p.n) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister
          p j odValue)).odRegisterValue =
    odValue
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister_odRegisterValue_eq
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j odValue : ℕ)
  (hod : odValue < 1 <<< p.n) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
        p
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister
          p j odValue)).odRegisterValue =
    odValue
    Splicing an n-bit value into the O_D block makes that value the extracted O_D register.
Theorem10.2.347
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper splice od register div high base eq of od value lt”; the hypotheses and conclusion in the code panel fix its exact scope. Splicing an n-bit O_D value preserves all bits above the O_D register.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Splicing an n-bit O_D value preserves all bits above the O_D register.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6439. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.347●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister_div_highBase_eq_of_odValue_lt
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j odValue : ℕ) (hod : odValue < 1 <<< p.n) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister p
        j odValue /
      2 ^ (1 + 2 * p.n) =
    j / 2 ^ (1 + 2 * p.n)
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister_div_highBase_eq_of_odValue_lt
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j odValue : ℕ)
  (hod : odValue < 1 <<< p.n) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister
        p j odValue /
      2 ^ (1 + 2 * p.n) =
    j / 2 ^ (1 + 2 * p.n)
    Splicing an n-bit O_D value preserves all bits above the O_D register.
Theorem10.2.348
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper splice od register lt qubit dim of od value lt”; the hypotheses and conclusion in the code panel fix its exact scope. Splicing an n-bit O_D value into a finite compound basis index preserves the full finite-basis range.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Splicing an n-bit O_D value into a finite compound basis index preserves the full finite-basis range. This is the range counterpart of the splice register equations and does not assert that the chosen O_D value is semantically correct.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6471. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.348●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister_lt_qubitDim_of_odValue_lt
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j odValue : ℕ)
  (hj :
    j <
      QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
  (hod : odValue < 1 <<< p.n) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister p
      j odValue <
    QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister_lt_qubitDim_of_odValue_lt
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j odValue : ℕ)
  (hj :
    j <
      QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p))
  (hod : odValue < 1 <<< p.n) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister
      p j odValue <
    QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p)
    Splicing an n-bit O_D value into a finite compound basis index preserves the
full finite-basis range.  This is the range counterpart of the splice register
equations and does not assert that the chosen O_D value is semantically correct.
Theorem10.2.349
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper splice od register splice of od value lt”; the hypotheses and conclusion in the code panel fix its exact scope. Replacing the O_D block twice is the same as keeping the second replacement.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Replacing the O_D block twice is the same as keeping the second replacement.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6515. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.349●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister_splice_of_odValue_lt
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j odValue newODValue : ℕ) (hod : odValue < 1 <<< p.n) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister p
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister
        p j odValue)
      newODValue =
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister
      p j newODValue
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister_splice_of_odValue_lt
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j odValue newODValue : ℕ)
  (hod : odValue < 1 <<< p.n) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister
      p
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister
        p j odValue)
      newODValue =
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister
      p j newODValue
    Replacing the O_D block twice is the same as keeping the second replacement.
Theorem10.2.350
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper splice od register self”; the hypotheses and conclusion in the code panel fix its exact scope. Reconstructing an index from its low, O_D, and high blocks gives the same index.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Reconstructing an index from its low, O_D, and high blocks gives the same index.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6530. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.350●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister_self
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister p
      j
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
          j).odRegisterValue =
    j
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister_self
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperSpliceODRegister
      p j
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
          p j).odRegisterValue =
    j
    Reconstructing an index from its low, O_D, and high blocks gives the same index.
Definition10.2.351
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access paper clean od value”. Clean 'O_D^BS' register value whose padded-low part is zero and sparse part is 'sparseValue'.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Clean 'O_D^BS' register value whose padded-low part is zero and sparse part is 'sparseValue'.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:6572. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.351●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanODValue
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (sparseValue : ℕ) : ℕ
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanODValue
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (sparseValue : ℕ) : ℕ
    Clean `O_D^BS` register value whose padded-low part is zero and sparse part is `sparseValue`.
Theorem10.2.352
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper clean od value padded zero eq zero”; the hypotheses and conclusion in the code panel fix its exact scope. The clean O_D value has zeroes in the padded low slice.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The clean O_D value has zeroes in the padded low slice.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6577. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.352●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanODValue_paddedZero_eq_zero
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (sparseValue : ℕ) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanODValue p
        sparseValue &&&
      1 <<< (p.n - QuantumBlockEncoding.clog2 p.kappa) - 1 =
    0
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanODValue_paddedZero_eq_zero
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (sparseValue : ℕ) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanODValue
        p sparseValue &&&
      1 <<<
          (p.n -
            QuantumBlockEncoding.clog2
              p.kappa) -
        1 =
    0
    The clean O_D value has zeroes in the padded low slice.
Theorem10.2.353
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper clean od value lt two pow of sparse lt”; the hypotheses and conclusion in the code panel fix its exact scope. A clean sparse value fits in the n-bit O_D register when the sparse width fits in n.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. A clean sparse value fits in the n-bit O_D register when the sparse width fits in n.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6586. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.353●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanODValue_lt_two_pow_of_sparse_lt
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  {sparseValue : ℕ} (hwidth : QuantumBlockEncoding.clog2 p.kappa ≤ p.n)
  (hsparse : sparseValue < 2 ^ QuantumBlockEncoding.clog2 p.kappa) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanODValue p
      sparseValue <
    1 <<< p.n
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanODValue_lt_two_pow_of_sparse_lt
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  {sparseValue : ℕ}
  (hwidth :
    QuantumBlockEncoding.clog2 p.kappa ≤
      p.n)
  (hsparse :
    sparseValue <
      2 ^
        QuantumBlockEncoding.clog2
          p.kappa) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanODValue
      p sparseValue <
    1 <<< p.n
    A clean sparse value fits in the n-bit O_D register when the sparse width fits in n.
Theorem10.2.354
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper clean od value sparse index eq”; the hypotheses and conclusion in the code panel fix its exact scope. Extracting the sparse slice from a clean O_D value recovers the sparse value.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Extracting the sparse slice from a clean O_D value recovers the sparse value.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6601. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.354●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanODValue_sparseIndex_eq
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  {sparseValue : ℕ}
  (hsparse : sparseValue < 2 ^ QuantumBlockEncoding.clog2 p.kappa) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanODValue p
          sparseValue >>>
        (p.n - QuantumBlockEncoding.clog2 p.kappa) &&&
      1 <<< QuantumBlockEncoding.clog2 p.kappa - 1 =
    sparseValue
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanODValue_sparseIndex_eq
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  {sparseValue : ℕ}
  (hsparse :
    sparseValue <
      2 ^
        QuantumBlockEncoding.clog2
          p.kappa) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanODValue
          p sparseValue >>>
        (p.n -
          QuantumBlockEncoding.clog2
            p.kappa) &&&
      1 <<<
          QuantumBlockEncoding.clog2
            p.kappa -
        1 =
    sparseValue
    Extracting the sparse slice from a clean O_D value recovers the sparse value.
Theorem10.2.355
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper clean input od register value eq clean od value”; the hypotheses and conclusion in the code panel fix its exact scope. On a clean Lemma 1 source column, the extracted O_D register is exactly the canonical clean sparse-register value for its sparse slot.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. On a clean Lemma 1 source column, the extracted O_D register is exactly the canonical clean sparse-register value for its sparse slot. This is the bit-slice reconstruction block needed to lift address injectivity to full paper-image injectivity. It only uses the executable clean-input predicate; it does not prove any semantic cleanup flag.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6621. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.355●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput_odRegisterValue_eq_cleanODValue
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ)
  (hwidth : QuantumBlockEncoding.clog2 p.kappa ≤ p.n)
  (hclean :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput p j =
      true) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
        j).odRegisterValue =
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanODValue p
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
          j).sparseIndexValue
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput_odRegisterValue_eq_cleanODValue
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ)
  (hwidth :
    QuantumBlockEncoding.clog2 p.kappa ≤
      p.n)
  (hclean :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput
        p j =
      true) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
        p j).odRegisterValue =
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanODValue
      p
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
          p j).sparseIndexValue
    On a clean Lemma 1 source column, the extracted O_D register is exactly the
canonical clean sparse-register value for its sparse slot.

This is the bit-slice reconstruction block needed to lift address injectivity
to full paper-image injectivity.  It only uses the executable clean-input
predicate; it does not prove any semantic cleanup flag.
Theorem10.2.356
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper image injective on global slot source”; the hypotheses and conclusion in the code panel fix its exact scope. The corrected active 'O_D^BS' paper image is injective on the faithful global-slot clean source domain.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The corrected active 'O_D^BS' paper image is injective on the faithful global-slot clean source domain. This is a finite-register proof-DAG block, not a semantic-flag promotion. It combines low-prefix preservation, high-tail preservation, the written-address roundtrip, same-row global-address injectivity, and clean O_D register reconstruction. The obligation records for inverse-on-range, dagger cleanup, and unitarity remain false.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6695. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.356●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage_injective_on_globalSlotSource
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j₁ j₂ :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits : QuantumBlockEncoding.clog2 p.kappa = 3)
  (hsource₁ :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑j₁ =
      true)
  (hsource₂ :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑j₂ =
      true)
  (himage :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p ↑j₁ =
      QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p ↑j₂) :
  j₁ = j₂
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage_injective_on_globalSlotSource
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j₁ j₂ :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits :
    QuantumBlockEncoding.clog2 p.kappa =
      3)
  (hsource₁ :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑j₁ =
      true)
  (hsource₂ :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑j₂ =
      true)
  (himage :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
        p ↑j₁ =
      QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
        p ↑j₂) :
  j₁ = j₂
    The corrected active `O_D^BS` paper image is injective on the faithful
global-slot clean source domain.

This is a finite-register proof-DAG block, not a semantic-flag promotion.  It
combines low-prefix preservation, high-tail preservation, the written-address
roundtrip, same-row global-address injectivity, and clean O_D register
reconstruction.  The obligation records for inverse-on-range, dagger cleanup,
and unitarity remain false.
Theorem10.2.357
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper post swap reverse sparse lt two pow”; the hypotheses and conclusion in the code panel fix its exact scope. The reverse sparse index used by the post-SWAP cleanup candidate fits in the three-bit sparse register for the one-term Robin parameter family.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The reverse sparse index used by the post-SWAP cleanup candidate fits in the three-bit sparse register for the one-term Robin parameter family.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6822. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.357●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapReverseSparse_lt_two_pow
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (source : ℕ)
  (hn : 3 ≤ p.n) (hκbits : QuantumBlockEncoding.clog2 p.kappa = 3) :
  QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
          source).sparseIndexValue <
    2 ^ QuantumBlockEncoding.clog2 p.kappa
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapReverseSparse_lt_two_pow
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source : ℕ) (hn : 3 ≤ p.n)
  (hκbits :
    QuantumBlockEncoding.clog2 p.kappa =
      3) :
  QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
          p source).sparseIndexValue <
    2 ^ QuantumBlockEncoding.clog2 p.kappa
    The reverse sparse index used by the post-SWAP cleanup candidate fits in the
three-bit sparse register for the one-term Robin parameter family.
Theorem10.2.358
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper post swap clean od value lt two pow”; the hypotheses and conclusion in the code panel fix its exact scope. The clean O_D register value spliced into the post-SWAP preimage candidate is n-bit for the one-term Robin parameter family.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The clean O_D register value spliced into the post-SWAP preimage candidate is n-bit for the one-term Robin parameter family.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6837. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.358●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapCleanODValue_lt_two_pow
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (source : ℕ)
  (hn : 3 ≤ p.n) (hκbits : QuantumBlockEncoding.clog2 p.kappa = 3) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanODValue p
      (QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
            source).sparseIndexValue) <
    1 <<< p.n
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapCleanODValue_lt_two_pow
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source : ℕ) (hn : 3 ≤ p.n)
  (hκbits :
    QuantumBlockEncoding.clog2 p.kappa =
      3) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanODValue
      p
      (QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
            p source).sparseIndexValue) <
    1 <<< p.n
    The clean O_D register value spliced into the post-SWAP preimage candidate is
n-bit for the one-term Robin parameter family.
Definition10.2.359
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access paper post swap preimage candidate”. Candidate clean preimage for the column reached by 'O_D^BS', SWAP, and then '(O_D^BS)^dagger'.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Candidate clean preimage for the column reached by 'O_D^BS', SWAP, and then '(O_D^BS)^dagger'. The candidate keeps the post-SWAP row and high-tail bits, and replaces the 'O_D^BS' register by a clean padded register whose sparse field is the inverse global slot for the original source slot. The separate Boolean audit below checks whether this candidate is actually a paper-image preimage.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:6865. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.359●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source : ℕ) : ℕ
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source : ℕ) : ℕ
    Candidate clean preimage for the column reached by
`O_D^BS`, SWAP, and then `(O_D^BS)^dagger`.

The candidate keeps the post-SWAP row and high-tail bits, and replaces the
`O_D^BS` register by a clean padded register whose sparse field is the inverse
global slot for the original source slot.  The separate Boolean audit below
checks whether this candidate is actually a paper-image preimage.
Definition10.2.360
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access paper post swap preimage candidate checks”. Executable audit for the post-SWAP preimage candidate.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Executable audit for the post-SWAP preimage candidate. It checks three local facts: the candidate maps by the active paper-image skeleton to the post-SWAP column, the candidate is in the clean padded domain, and the candidate address is n-bit. Even when this Boolean is true for a finite parameter scan, the paper-level dagger cleanup and unitarity flags remain unproved.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:6882. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.360●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidateChecks
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source : ℕ) : Bool
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidateChecks
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source : ℕ) : Bool
    Executable audit for the post-SWAP preimage candidate.

It checks three local facts: the candidate maps by the active paper-image
skeleton to the post-SWAP column, the candidate is in the clean padded domain,
and the candidate address is n-bit.  Even when this Boolean is true for a
finite parameter scan, the paper-level dagger cleanup and unitarity flags
remain unproved.
Theorem10.2.361
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper post swap preimage candidate checks of clean source”; the hypotheses and conclusion in the code panel fix its exact scope. The post-SWAP preimage candidate passes the executable image, clean-domain, and address-range checks for clean one-term Robin source columns.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The post-SWAP preimage candidate passes the executable image, clean-domain, and address-range checks for clean one-term Robin source columns. The assumptions keep the current Phase 1 contract explicit: the source column is in the finite basis, the source padded register is clean, and the one-term family uses a three-bit sparse register ('kappa = 7', 'clog2 kappa = 3'). This does not prove uniqueness, dagger cleanup, unitarity, or block extraction.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:6899. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.361●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidateChecks_of_cleanSource
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (source : ℕ)
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits : QuantumBlockEncoding.clog2 p.kappa = 3)
  (hsource :
    source <
      QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
  (hclean :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput p
        source =
      true) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidateChecks
      p source =
    true
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidateChecks_of_cleanSource
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source : ℕ) (hn : 3 ≤ p.n)
  (hkappa : p.kappa = 7)
  (hκbits :
    QuantumBlockEncoding.clog2 p.kappa =
      3)
  (hsource :
    source <
      QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p))
  (hclean :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput
        p source =
      true) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidateChecks
      p source =
    true
    The post-SWAP preimage candidate passes the executable image, clean-domain, and
address-range checks for clean one-term Robin source columns.

The assumptions keep the current Phase 1 contract explicit: the source column is
in the finite basis, the source padded register is clean, and the one-term
family uses a three-bit sparse register (`kappa = 7`, `clog2 kappa = 3`).  This
does not prove uniqueness, dagger cleanup, unitarity, or block extraction.
Theorem10.2.362
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper post swap preimage candidate lt qubit dim of clean source”; the hypotheses and conclusion in the code panel fix its exact scope. The clean post-SWAP preimage candidate is a finite basis index for finite clean one-term Robin source columns.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The clean post-SWAP preimage candidate is a finite basis index for finite clean one-term Robin source columns. This only discharges the 'Fin' constructor premise for the conditional cleanup witness; uniqueness and semantic cleanup remain open.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:7006. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.362●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate_lt_qubitDim_of_cleanSource
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits : QuantumBlockEncoding.clog2 p.kappa = 3)
  (hclean :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput p
        ↑source =
      true) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate
      p ↑source <
    QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate_lt_qubitDim_of_cleanSource
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits :
    QuantumBlockEncoding.clog2 p.kappa =
      3)
  (hclean :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput
        p ↑source =
      true) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate
      p ↑source <
    QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p)
    The clean post-SWAP preimage candidate is a finite basis index for finite clean
one-term Robin source columns.  This only discharges the `Fin` constructor
premise for the conditional cleanup witness; uniqueness and semantic cleanup
remain open.
Theorem10.2.363
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access post swap cleanup of clean source candidate”; the hypotheses and conclusion in the code panel fix its exact scope. Instantiate the conditional post-SWAP cleanup witness with the clean-source preimage candidate.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Instantiate the conditional post-SWAP cleanup witness with the clean-source preimage candidate. The finite 'post' and 'pre' range facts remain explicit hypotheses. This wrapper converts the accepted Boolean candidate audit into the supplied preimage equality, clean-domain proof, and n-bit address bound required by 'bandedSparseAccessPostSwapCleanup_of_preimage'. It does not prove finite range, uniqueness, semantic dagger cleanup, or unitarity.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:7055. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.363●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPostSwapCleanup_of_cleanSourceCandidate
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits : QuantumBlockEncoding.clog2 p.kappa = 3)
  (hclean :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput p
        ↑source =
      true)
  (hpostRange :
    QuantumBlockEncoding.GHL2025.swapOracleImage p
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p
          ↑source) <
      QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
  (hpreRange :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate
        p ↑source <
      QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)) :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessPostSwapCleanup p
    source
    ⟨QuantumBlockEncoding.GHL2025.swapOracleImage p
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p
          ↑source),
      hpostRange⟩
    ⟨QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate
        p ↑source,
      hpreRange⟩
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPostSwapCleanup_of_cleanSourceCandidate
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits :
    QuantumBlockEncoding.clog2 p.kappa =
      3)
  (hclean :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput
        p ↑source =
      true)
  (hpostRange :
    QuantumBlockEncoding.GHL2025.swapOracleImage
        p
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
          p ↑source) <
      QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p))
  (hpreRange :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate
        p ↑source <
      QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)) :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessPostSwapCleanup
    p source
    ⟨QuantumBlockEncoding.GHL2025.swapOracleImage
        p
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
          p ↑source),
      hpostRange⟩
    ⟨QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate
        p ↑source,
      hpreRange⟩
    Instantiate the conditional post-SWAP cleanup witness with the clean-source
preimage candidate.

The finite `post` and `pre` range facts remain explicit hypotheses.  This
wrapper converts the accepted Boolean candidate audit into the supplied
preimage equality, clean-domain proof, and n-bit address bound required by
`bandedSparseAccessPostSwapCleanup_of_preimage`.  It does not prove finite
range, uniqueness, semantic dagger cleanup, or unitarity.
Theorem10.2.364
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access post swap cleanup of clean source candidate no range”; the hypotheses and conclusion in the code panel fix its exact scope. Instantiate the clean-source post-SWAP cleanup witness without caller-supplied finite-range premises.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Instantiate the clean-source post-SWAP cleanup witness without caller-supplied finite-range premises. The theorem only supplies the 'Fin' range proofs for the already conditional candidate witness; it does not prove uniqueness, dagger cleanup, or either O_D^BS unitarity flag.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:7096. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.364●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPostSwapCleanup_of_cleanSourceCandidate_noRange
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits : QuantumBlockEncoding.clog2 p.kappa = 3)
  (hclean :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput p
        ↑source =
      true) :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessPostSwapCleanup p
    source
    ⟨QuantumBlockEncoding.GHL2025.swapOracleImage p
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p
          ↑source),
      ⋯⟩
    ⟨QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate
        p ↑source,
      ⋯⟩
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPostSwapCleanup_of_cleanSourceCandidate_noRange
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits :
    QuantumBlockEncoding.clog2 p.kappa =
      3)
  (hclean :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperCleanInput
        p ↑source =
      true) :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessPostSwapCleanup
    p source
    ⟨QuantumBlockEncoding.GHL2025.swapOracleImage
        p
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
          p ↑source),
      ⋯⟩
    ⟨QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate
        p ↑source,
      ⋯⟩
    Instantiate the clean-source post-SWAP cleanup witness without caller-supplied
finite-range premises.  The theorem only supplies the `Fin` range proofs for
the already conditional candidate witness; it does not prove uniqueness,
dagger cleanup, or either O_D^BS unitarity flag.
Theorem10.2.365
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access post swap cleanup of valid clean source candidate no range”; the hypotheses and conclusion in the code panel fix its exact scope. Feed the row-dependent valid-clean-source predicate into the existing post-SWAP cleanup candidate wrapper.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Feed the row-dependent valid-clean-source predicate into the existing post-SWAP cleanup candidate wrapper. The predicate 'bandedSparseAccessPaperValidCleanSource' is only a Phase 1 source-domain classifier. This theorem records that it supplies the clean padded-register hypothesis required by the cleanup candidate. It does not prove source-domain completeness, unused-branch unitary extension, preimage uniqueness, semantic dagger cleanup, or either O_D^BS unitarity flag.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:7144. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.365●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPostSwapCleanup_of_validCleanSourceCandidate_noRange
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits : QuantumBlockEncoding.clog2 p.kappa = 3)
  (hvalid :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource
        p ↑source =
      true) :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessPostSwapCleanup p
    source
    ⟨QuantumBlockEncoding.GHL2025.swapOracleImage p
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p
          ↑source),
      ⋯⟩
    ⟨QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate
        p ↑source,
      ⋯⟩
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPostSwapCleanup_of_validCleanSourceCandidate_noRange
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits :
    QuantumBlockEncoding.clog2 p.kappa =
      3)
  (hvalid :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource
        p ↑source =
      true) :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessPostSwapCleanup
    p source
    ⟨QuantumBlockEncoding.GHL2025.swapOracleImage
        p
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
          p ↑source),
      ⋯⟩
    ⟨QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate
        p ↑source,
      ⋯⟩
    Feed the row-dependent valid-clean-source predicate into the existing
post-SWAP cleanup candidate wrapper.

The predicate `bandedSparseAccessPaperValidCleanSource` is only a Phase 1
source-domain classifier.  This theorem records that it supplies the clean
padded-register hypothesis required by the cleanup candidate.  It does not
prove source-domain completeness, unused-branch unitary extension, preimage
uniqueness, semantic dagger cleanup, or either O_D^BS unitarity flag.
Theorem10.2.366
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access post swap cleanup of global slot source candidate no range”; the hypotheses and conclusion in the code panel fix its exact scope. Feed the faithful global-slot source predicate into the existing post-SWAP cleanup candidate wrapper.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Feed the faithful global-slot source predicate into the existing post-SWAP cleanup candidate wrapper. This is the active-source analogue of the row-dependent valid-clean-source bridge. It only extracts the padded clean-input fact from 'bandedSparseAccessPaperGlobalSlotSource' and reuses the conditional cleanup candidate. It does not prove preimage uniqueness, semantic dagger cleanup, unitarity, LCU correctness, or block extraction.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:7186. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.366●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPostSwapCleanup_of_globalSlotSourceCandidate_noRange
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits : QuantumBlockEncoding.clog2 p.kappa = 3)
  (hsource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑source =
      true) :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessPostSwapCleanup p
    source
    ⟨QuantumBlockEncoding.GHL2025.swapOracleImage p
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p
          ↑source),
      ⋯⟩
    ⟨QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate
        p ↑source,
      ⋯⟩
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPostSwapCleanup_of_globalSlotSourceCandidate_noRange
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits :
    QuantumBlockEncoding.clog2 p.kappa =
      3)
  (hsource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑source =
      true) :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessPostSwapCleanup
    p source
    ⟨QuantumBlockEncoding.GHL2025.swapOracleImage
        p
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
          p ↑source),
      ⋯⟩
    ⟨QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate
        p ↑source,
      ⋯⟩
    Feed the faithful global-slot source predicate into the existing post-SWAP
cleanup candidate wrapper.

This is the active-source analogue of the row-dependent valid-clean-source
bridge.  It only extracts the padded clean-input fact from
`bandedSparseAccessPaperGlobalSlotSource` and reuses the conditional cleanup
candidate.  It does not prove preimage uniqueness, semantic dagger cleanup,
unitarity, LCU correctness, or block extraction.
Theorem10.2.367
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper post swap preimage candidate checks of global slot source”; the hypotheses and conclusion in the code panel fix its exact scope. The post-SWAP preimage candidate audit is available on the active global-slot source domain.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The post-SWAP preimage candidate audit is available on the active global-slot source domain. This is the global-source wrapper around the existing clean-source arithmetic block. It proves only the executable candidate check; inverse uniqueness, semantic dagger cleanup, and unitarity remain separate obligations.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:7226. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.367●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidateChecks_of_globalSlotSource
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (source : ℕ)
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits : QuantumBlockEncoding.clog2 p.kappa = 3)
  (hsourceFinite :
    source <
      QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
  (hsource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p source =
      true) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidateChecks
      p source =
    true
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidateChecks_of_globalSlotSource
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source : ℕ) (hn : 3 ≤ p.n)
  (hkappa : p.kappa = 7)
  (hκbits :
    QuantumBlockEncoding.clog2 p.kappa =
      3)
  (hsourceFinite :
    source <
      QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p))
  (hsource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p source =
      true) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidateChecks
      p source =
    true
    The post-SWAP preimage candidate audit is available on the active global-slot
source domain.

This is the global-source wrapper around the existing clean-source arithmetic
block.  It proves only the executable candidate check; inverse uniqueness,
semantic dagger cleanup, and unitarity remain separate obligations.
Theorem10.2.368
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper post swap preimage candidate lt qubit dim of global slot source”; the hypotheses and conclusion in the code panel fix its exact scope. The global-source post-SWAP preimage candidate is a finite basis index.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The global-source post-SWAP preimage candidate is a finite basis index. This theorem only names the range premise needed by the conditional cleanup witness. It does not prove that the candidate is unique or that the dagger cleans every in-range image.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:7245. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.368●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate_lt_qubitDim_of_globalSlotSource
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits : QuantumBlockEncoding.clog2 p.kappa = 3)
  (hsource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑source =
      true) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate
      p ↑source <
    QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate_lt_qubitDim_of_globalSlotSource
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits :
    QuantumBlockEncoding.clog2 p.kappa =
      3)
  (hsource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑source =
      true) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate
      p ↑source <
    QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p)
    The global-source post-SWAP preimage candidate is a finite basis index.

This theorem only names the range premise needed by the conditional cleanup
witness.  It does not prove that the candidate is unique or that the dagger
cleans every in-range image.
Theorem10.2.369
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper post swap preimage candidate sparse index eq”; the hypotheses and conclusion in the code panel fix its exact scope. The post-SWAP preimage candidate has the reverse sparse slot in its extracted clean O_D register.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The post-SWAP preimage candidate has the reverse sparse slot in its extracted clean O_D register. This names the splice/sparse-slice calculation used by the cleanup-candidate audit so the unique-preimage route can reuse it instead of repeating the bit-level proof.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:7267. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.369●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate_sparseIndex_eq
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (source : ℕ)
  (hn : 3 ≤ p.n) (hκbits : QuantumBlockEncoding.clog2 p.kappa = 3) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate
          p source)).sparseIndexValue =
    QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters p
          source).sparseIndexValue
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate_sparseIndex_eq
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source : ℕ) (hn : 3 ≤ p.n)
  (hκbits :
    QuantumBlockEncoding.clog2 p.kappa =
      3) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
        p
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate
          p source)).sparseIndexValue =
    QuantumBlockEncoding.GHL2025.oneTermRobinGlobalSparseInverseSlot
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperRegisters
          p source).sparseIndexValue
    The post-SWAP preimage candidate has the reverse sparse slot in its extracted
clean O_D register.

This names the splice/sparse-slice calculation used by the cleanup-candidate
audit so the unique-preimage route can reuse it instead of repeating the
bit-level proof.
Theorem10.2.370
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper post swap preimage candidate global slot source of global slot source”; the hypotheses and conclusion in the code panel fix its exact scope. The post-SWAP preimage candidate is itself an active global-slot source.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The post-SWAP preimage candidate is itself an active global-slot source. The proof combines the executable candidate audit for clean input with the named reverse-slot sparse-slice lemma above. It still does not promote the inverse-on-range, uniqueness, dagger-cleanup, or unitarity obligation flags.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:7312. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.370●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate_globalSlotSource_of_globalSlotSource
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits : QuantumBlockEncoding.clog2 p.kappa = 3)
  (hsource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑source =
      true) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource p
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate
        p ↑source) =
    true
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate_globalSlotSource_of_globalSlotSource
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits :
    QuantumBlockEncoding.clog2 p.kappa =
      3)
  (hsource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑source =
      true) :
  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
      p
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate
        p ↑source) =
    true
    The post-SWAP preimage candidate is itself an active global-slot source.

The proof combines the executable candidate audit for clean input with the
named reverse-slot sparse-slice lemma above.  It still does not promote the
inverse-on-range, uniqueness, dagger-cleanup, or unitarity obligation flags.
Theorem10.2.371
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access paper post swap preimage candidate unique on global slot source”; the hypotheses and conclusion in the code panel fix its exact scope. Uniqueness of the active global-slot clean preimage for the post-SWAP target.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Uniqueness of the active global-slot clean preimage for the post-SWAP target. Any active global-source column that maps by the corrected paper image to the post-SWAP target must be the named reverse-slot preimage candidate. This is a finite basis-index lemma for the dagger-cleanup route; semantic obligation records remain false.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:7360. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.371●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate_unique_on_globalSlotSource
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source pre :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits : QuantumBlockEncoding.clog2 p.kappa = 3)
  (hsource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑source =
      true)
  (hpreSource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑pre =
      true)
  (hpreImage :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p ↑pre =
      QuantumBlockEncoding.GHL2025.swapOracleImage p
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p
          ↑source)) :
  ↑pre =
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate
      p ↑source
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate_unique_on_globalSlotSource
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source pre :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits :
    QuantumBlockEncoding.clog2 p.kappa =
      3)
  (hsource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑source =
      true)
  (hpreSource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑pre =
      true)
  (hpreImage :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
        p ↑pre =
      QuantumBlockEncoding.GHL2025.swapOracleImage
        p
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
          p ↑source)) :
  ↑pre =
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperPostSwapPreimageCandidate
      p ↑source
    Uniqueness of the active global-slot clean preimage for the post-SWAP target.

Any active global-source column that maps by the corrected paper image to the
post-SWAP target must be the named reverse-slot preimage candidate.  This is a
finite basis-index lemma for the dagger-cleanup route; semantic obligation
records remain false.
Definition10.2.372
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “banded sparse access global slot inverse on range contract”. A proposition-valued field is a requirement until a constructor supplies it. Proof-obligation interface for the active global-slot inverse-on-range route.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Proof-obligation interface for the active global-slot inverse-on-range route. The record fixes the source predicate, image function, post-SWAP target, and candidate preimage used by the corrected 'O_D^BS' route. The executable 'candidateChecks' field can be proved from 'bandedSparseAccessPaperGlobalSlotSource' by 'bandedSparseAccessPaperPostSwapPreimageCandidateChecks_of_globalSlotSource'. The semantic inverse, uniqueness, injectivity, cleanup, and unitary-extension fields remain false in Phase 1.

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:7419. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.372●1 definition
  • structure(15 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.BandedSparseAccessGlobalSlotInverseOnRangeContract :
  Type
    structure QuantumBlockEncoding.GHL2025.BandedSparseAccessGlobalSlotInverseOnRangeContract :
  Type
    Proof-obligation interface for the active global-slot inverse-on-range route.

The record fixes the source predicate, image function, post-SWAP target, and
candidate preimage used by the corrected `O_D^BS` route.  The executable
`candidateChecks` field can be proved from
`bandedSparseAccessPaperGlobalSlotSource` by
`bandedSparseAccessPaperPostSwapPreimageCandidateChecks_of_globalSlotSource`.
The semantic inverse, uniqueness, injectivity, cleanup, and unitary-extension
fields remain false in Phase 1.

    Fields

    sourceAnchor : String
    sourcePredicate : String
    imageFunction : String
    preimageCandidate : String
    sourceIndex : ℕ
    sourceRegisters : QuantumBlockEncoding.GHL2025.BandedSparseAccessPaperRegisters
    sourceInGlobalDomain : Bool
    postSwapImageIndex : ℕ
    candidatePreimageIndex : ℕ
    candidateChecks : Bool
    inverseOnRange : QuantumBlockEncoding.GHL2025.ObligationRecord
    uniquePreimage : QuantumBlockEncoding.GHL2025.ObligationRecord
    imageInjectiveOnGlobalSource : QuantumBlockEncoding.GHL2025.ObligationRecord
    daggerCleanup : QuantumBlockEncoding.GHL2025.ObligationRecord
    unitaryExtension : QuantumBlockEncoding.GHL2025.ObligationRecord
Definition10.2.373
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access global slot inverse on range contract”. Default global-source inverse-on-range contract for one 'O_D^BS' source column.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Default global-source inverse-on-range contract for one 'O_D^BS' source column. This records the fixed route for future proof work without promoting any semantic flag or changing the active forward/dagger matrices.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:7443. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.373●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source : ℕ) :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessGlobalSlotInverseOnRangeContract
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source : ℕ) :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessGlobalSlotInverseOnRangeContract
    Default global-source inverse-on-range contract for one `O_D^BS` source column.

This records the fixed route for future proof work without promoting any
semantic flag or changing the active forward/dagger matrices.
Theorem10.2.374
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access global slot inverse on range contract flags false”; the hypotheses and conclusion in the code panel fix its exact scope. The global-source inverse-on-range contract is obligation-only in Phase 1.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The global-source inverse-on-range contract is obligation-only in Phase 1.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:7485. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.374●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract_flags_false
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
            p source).inverseOnRange.proved =
      false ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
              p source).uniquePreimage.proved =
        false ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                p source).imageInjectiveOnGlobalSource.proved =
          false ∧
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                  p source).daggerCleanup.proved =
            false ∧
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                  p source).unitaryExtension.proved =
            false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract_flags_false
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
            p
            source).inverseOnRange.proved =
      false ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
              p
              source).uniquePreimage.proved =
        false ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                p
                source).imageInjectiveOnGlobalSource.proved =
          false ∧
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                  p
                  source).daggerCleanup.proved =
            false ∧
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                  p
                  source).unitaryExtension.proved =
            false
    The global-source inverse-on-range contract is obligation-only in Phase 1.
Theorem10.2.375
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access global slot inverse on range contract of global slot source”; the hypotheses and conclusion in the code panel fix its exact scope. Global-source columns feed the fixed inverse-on-range interface and satisfy the executable candidate audit.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Global-source columns feed the fixed inverse-on-range interface and satisfy the executable candidate audit. The final fields remain false: this theorem does not assert uniqueness, injectivity, semantic dagger cleanup, or unitarity.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:7506. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.375●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract_of_globalSlotSource
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (source : ℕ)
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits : QuantumBlockEncoding.clog2 p.kappa = 3)
  (hsourceFinite :
    source <
      QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))
  (hsource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p source =
      true) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
          p source).sourceInGlobalDomain =
      true ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
            p source).candidateChecks =
        true ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                p source).inverseOnRange.proved =
          false ∧
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                  p source).uniquePreimage.proved =
            false ∧
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                    p source).imageInjectiveOnGlobalSource.proved =
              false ∧
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                      p source).daggerCleanup.proved =
                false ∧
              (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                      p source).unitaryExtension.proved =
                false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract_of_globalSlotSource
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source : ℕ) (hn : 3 ≤ p.n)
  (hkappa : p.kappa = 7)
  (hκbits :
    QuantumBlockEncoding.clog2 p.kappa =
      3)
  (hsourceFinite :
    source <
      QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p))
  (hsource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p source =
      true) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
          p source).sourceInGlobalDomain =
      true ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
            p source).candidateChecks =
        true ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                p
                source).inverseOnRange.proved =
          false ∧
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                  p
                  source).uniquePreimage.proved =
            false ∧
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                    p
                    source).imageInjectiveOnGlobalSource.proved =
              false ∧
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                      p
                      source).daggerCleanup.proved =
                false ∧
              (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                      p
                      source).unitaryExtension.proved =
                false
    Global-source columns feed the fixed inverse-on-range interface and satisfy the
executable candidate audit.

The final fields remain false: this theorem does not assert uniqueness,
injectivity, semantic dagger cleanup, or unitarity.
Theorem10.2.376
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access global slot inverse on range contract unique preimage bridge”; the hypotheses and conclusion in the code panel fix its exact scope. Record-level bridge from the compiled post-SWAP unique-preimage theorem to the global-slot inverse-on-range contract.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Record-level bridge from the compiled post-SWAP unique-preimage theorem to the global-slot inverse-on-range contract. This theorem reflects the finite basis-index evidence in the contract fields: any active global-source preimage of the contract's post-SWAP target is the contract's candidate preimage. It deliberately keeps every semantic obligation flag in the contract false; dagger cleanup and unitarity remain separate Phase 1 obligations.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:7541. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.376●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract_uniquePreimageBridge
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source pre :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits : QuantumBlockEncoding.clog2 p.kappa = 3)
  (hsource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑source =
      true)
  (hpreSource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑pre =
      true)
  (hpreImage :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage p ↑pre =
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
          p ↑source).postSwapImageIndex) :
  ↑pre =
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
          p ↑source).candidatePreimageIndex ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
            p ↑source).sourceInGlobalDomain =
        true ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
              p ↑source).candidateChecks =
          true ∧
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                  p ↑source).inverseOnRange.proved =
            false ∧
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                    p ↑source).uniquePreimage.proved =
              false ∧
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                      p ↑source).imageInjectiveOnGlobalSource.proved =
                false ∧
              (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                        p ↑source).daggerCleanup.proved =
                  false ∧
                (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                        p ↑source).unitaryExtension.proved =
                  false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract_uniquePreimageBridge
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source pre :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits :
    QuantumBlockEncoding.clog2 p.kappa =
      3)
  (hsource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑source =
      true)
  (hpreSource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑pre =
      true)
  (hpreImage :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
        p ↑pre =
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
          p ↑source).postSwapImageIndex) :
  ↑pre =
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
          p
          ↑source).candidatePreimageIndex ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
            p
            ↑source).sourceInGlobalDomain =
        true ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
              p ↑source).candidateChecks =
          true ∧
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                  p
                  ↑source).inverseOnRange.proved =
            false ∧
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                    p
                    ↑source).uniquePreimage.proved =
              false ∧
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                      p
                      ↑source).imageInjectiveOnGlobalSource.proved =
                false ∧
              (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                        p
                        ↑source).daggerCleanup.proved =
                  false ∧
                (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                        p
                        ↑source).unitaryExtension.proved =
                  false
    Record-level bridge from the compiled post-SWAP unique-preimage theorem to the
global-slot inverse-on-range contract.

This theorem reflects the finite basis-index evidence in the contract fields:
any active global-source preimage of the contract's post-SWAP target is the
contract's candidate preimage.  It deliberately keeps every semantic obligation
flag in the contract false; dagger cleanup and unitarity remain separate
Phase 1 obligations.
Theorem10.2.377
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access global slot inverse on range contract dagger cleanup bridge”; the hypotheses and conclusion in the code panel fix its exact scope. Bridge the global-slot inverse-on-range contract to the concrete post-SWAP dagger cleanup witness.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Bridge the global-slot inverse-on-range contract to the concrete post-SWAP dagger cleanup witness. This theorem constructs the finite post-SWAP column and the named preimage candidate from the active global-source route, then reuses 'BandedSparseAccessPostSwapCleanup' to expose the transpose-style dagger entry and executable register checks. It deliberately keeps the semantic 'daggerCleanup' and 'unitaryExtension' flags false.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:7608. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.377●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract_daggerCleanupBridge
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits : QuantumBlockEncoding.clog2 p.kappa = 3)
  (hsource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑source =
      true) :
  ∃ post pre,
    QuantumBlockEncoding.GHL2025.BandedSparseAccessPostSwapCleanup p
        source post pre ∧
      ↑post =
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
              p ↑source).postSwapImageIndex ∧
        ↑pre =
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                p ↑source).candidatePreimageIndex ∧
          (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
                    p).matrix
                pre post =
              QuantumBlockEncoding.Coeff.rat 1 ∧
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                    p ↑source).candidateChecks =
                true ∧
              (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                        p ↑source).daggerCleanup.proved =
                  false ∧
                (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                        p ↑source).unitaryExtension.proved =
                  false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract_daggerCleanupBridge
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits :
    QuantumBlockEncoding.clog2 p.kappa =
      3)
  (hsource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑source =
      true) :
  ∃ post pre,
    QuantumBlockEncoding.GHL2025.BandedSparseAccessPostSwapCleanup
        p source post pre ∧
      ↑post =
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
              p
              ↑source).postSwapImageIndex ∧
        ↑pre =
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                p
                ↑source).candidatePreimageIndex ∧
          (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
                    p).matrix
                pre post =
              QuantumBlockEncoding.Coeff.rat
                1 ∧
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                    p
                    ↑source).candidateChecks =
                true ∧
              (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                        p
                        ↑source).daggerCleanup.proved =
                  false ∧
                (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                        p
                        ↑source).unitaryExtension.proved =
                  false
    Bridge the global-slot inverse-on-range contract to the concrete post-SWAP
dagger cleanup witness.

This theorem constructs the finite post-SWAP column and the named preimage
candidate from the active global-source route, then reuses
`BandedSparseAccessPostSwapCleanup` to expose the transpose-style dagger entry
and executable register checks.  It deliberately keeps the semantic
`daggerCleanup` and `unitaryExtension` flags false.
Theorem10.2.378
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access global slot inverse on range contract cleanup contract map”; the hypotheses and conclusion in the code panel fix its exact scope. Reviewed cleanup-contract map for the active global-slot 'O_D^BS' route.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Reviewed cleanup-contract map for the active global-slot 'O_D^BS' route. This wrapper is intentionally non-promoting. It combines the compiled post-SWAP cleanup witness with the record-level unique-preimage bridge, so later cleanup work can consume one theorem exposing the contract target, candidate preimage, active-source uniqueness, and transpose-style dagger entry. The semantic inverse, uniqueness, injectivity, cleanup, and unitary-extension obligation flags remain false.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:7681. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.378●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract_cleanupContractMap
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits : QuantumBlockEncoding.clog2 p.kappa = 3)
  (hsource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑source =
      true) :
  ∃ post pre,
    QuantumBlockEncoding.GHL2025.BandedSparseAccessPostSwapCleanup p
        source post pre ∧
      ↑post =
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
              p ↑source).postSwapImageIndex ∧
        ↑pre =
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                p ↑source).candidatePreimageIndex ∧
          QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
                p ↑pre =
              true ∧
            (∀
                (pre' :
                  Fin
                    (QuantumBlockEncoding.qubitDim
                      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
                        p))),
                QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
                      p ↑pre' =
                    true →
                  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
                        p ↑pre' =
                      ↑post →
                    ↑pre' = ↑pre) ∧
              (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
                        p).matrix
                    pre post =
                  QuantumBlockEncoding.Coeff.rat 1 ∧
                (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                        p ↑source).candidateChecks =
                    true ∧
                  (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                            p ↑source).inverseOnRange.proved =
                      false ∧
                    (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                              p ↑source).uniquePreimage.proved =
                        false ∧
                      (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                                p
                                ↑source).imageInjectiveOnGlobalSource.proved =
                          false ∧
                        (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                                  p ↑source).daggerCleanup.proved =
                            false ∧
                          (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                                  p ↑source).unitaryExtension.proved =
                            false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract_cleanupContractMap
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits :
    QuantumBlockEncoding.clog2 p.kappa =
      3)
  (hsource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑source =
      true) :
  ∃ post pre,
    QuantumBlockEncoding.GHL2025.BandedSparseAccessPostSwapCleanup
        p source post pre ∧
      ↑post =
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
              p
              ↑source).postSwapImageIndex ∧
        ↑pre =
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                p
                ↑source).candidatePreimageIndex ∧
          QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
                p ↑pre =
              true ∧
            (∀
                (pre' :
                  Fin
                    (QuantumBlockEncoding.qubitDim
                      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
                        p))),
                QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
                      p ↑pre' =
                    true →
                  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
                        p ↑pre' =
                      ↑post →
                    ↑pre' = ↑pre) ∧
              (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
                        p).matrix
                    pre post =
                  QuantumBlockEncoding.Coeff.rat
                    1 ∧
                (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                        p
                        ↑source).candidateChecks =
                    true ∧
                  (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                            p
                            ↑source).inverseOnRange.proved =
                      false ∧
                    (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                              p
                              ↑source).uniquePreimage.proved =
                        false ∧
                      (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                                p
                                ↑source).imageInjectiveOnGlobalSource.proved =
                          false ∧
                        (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                                  p
                                  ↑source).daggerCleanup.proved =
                            false ∧
                          (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                                  p
                                  ↑source).unitaryExtension.proved =
                            false
    Reviewed cleanup-contract map for the active global-slot `O_D^BS` route.

This wrapper is intentionally non-promoting.  It combines the compiled
post-SWAP cleanup witness with the record-level unique-preimage bridge, so
later cleanup work can consume one theorem exposing the contract target,
candidate preimage, active-source uniqueness, and transpose-style dagger entry.
The semantic inverse, uniqueness, injectivity, cleanup, and unitary-extension
obligation flags remain false.
Theorem10.2.379
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “default banded sparse access paper contract cleanup route bridge”; the hypotheses and conclusion in the code panel fix its exact scope. Default-paper-contract cleanup-route bridge for the active global-slot 'O_D^BS' route.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Default-paper-contract cleanup-route bridge for the active global-slot 'O_D^BS' route. This theorem ties the compiled cleanup-contract map back to 'defaultBandedSparseAccessPaperContract p'. It exposes the post-SWAP cleanup witness, active-source uniqueness, and transpose-style dagger entry while recording that the paper-contract cleanup and unitary-extension flags, and the two active 'O_D^BS' gate-unitarity flags, remain false.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:7775. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.379●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract_cleanupRouteBridge
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits : QuantumBlockEncoding.clog2 p.kappa = 3)
  (hsource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑source =
      true) :
  ∃ post pre,
    QuantumBlockEncoding.GHL2025.BandedSparseAccessPostSwapCleanup p
        source post pre ∧
      ↑post =
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
              p ↑source).postSwapImageIndex ∧
        ↑pre =
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                p ↑source).candidatePreimageIndex ∧
          QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
                p ↑pre =
              true ∧
            (∀
                (pre' :
                  Fin
                    (QuantumBlockEncoding.qubitDim
                      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
                        p))),
                QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
                      p ↑pre' =
                    true →
                  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
                        p ↑pre' =
                      ↑post →
                    ↑pre' = ↑pre) ∧
              (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
                        p).matrix
                    pre post =
                  QuantumBlockEncoding.Coeff.rat 1 ∧
                (QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract
                          p).daggerCleanup.proved =
                    false ∧
                  (QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract
                            p).unitaryExtension.proved =
                      false ∧
                    (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
                              p).unitary.proved =
                        false ∧
                      (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
                              p).unitary.proved =
                        false
    theorem QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract_cleanupRouteBridge
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits :
    QuantumBlockEncoding.clog2 p.kappa =
      3)
  (hsource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑source =
      true) :
  ∃ post pre,
    QuantumBlockEncoding.GHL2025.BandedSparseAccessPostSwapCleanup
        p source post pre ∧
      ↑post =
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
              p
              ↑source).postSwapImageIndex ∧
        ↑pre =
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                p
                ↑source).candidatePreimageIndex ∧
          QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
                p ↑pre =
              true ∧
            (∀
                (pre' :
                  Fin
                    (QuantumBlockEncoding.qubitDim
                      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
                        p))),
                QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
                      p ↑pre' =
                    true →
                  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
                        p ↑pre' =
                      ↑post →
                    ↑pre' = ↑pre) ∧
              (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
                        p).matrix
                    pre post =
                  QuantumBlockEncoding.Coeff.rat
                    1 ∧
                (QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract
                          p).daggerCleanup.proved =
                    false ∧
                  (QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract
                            p).unitaryExtension.proved =
                      false ∧
                    (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
                              p).unitary.proved =
                        false ∧
                      (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
                              p).unitary.proved =
                        false
    Default-paper-contract cleanup-route bridge for the active global-slot
`O_D^BS` route.

This theorem ties the compiled cleanup-contract map back to
`defaultBandedSparseAccessPaperContract p`.  It exposes the post-SWAP cleanup
witness, active-source uniqueness, and transpose-style dagger entry while
recording that the paper-contract cleanup and unitary-extension flags, and the
two active `O_D^BS` gate-unitarity flags, remain false.
Theorem10.2.380
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access global slot inverse on range contract dagger off candidate zero”; the hypotheses and conclusion in the code panel fix its exact scope. Off-candidate dagger entries are zero on the active global-slot source domain.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Off-candidate dagger entries are zero on the active global-slot source domain. For the fixed post-SWAP target in the global-slot inverse-on-range contract, any other active global-source preimage whose index is not the named candidate cannot have a transpose-style '(O_D^BS)^†' entry into that target. This is a matrix-entry bridge only: it does not promote dagger cleanup or unitary flags.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:7820. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.380●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract_daggerOffCandidate_zero
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source other post :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits : QuantumBlockEncoding.clog2 p.kappa = 3)
  (hsource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑source =
      true)
  (hotherSource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑other =
      true)
  (hpost :
    ↑post =
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
          p ↑source).postSwapImageIndex)
  (hne :
    ↑other ≠
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
          p ↑source).candidatePreimageIndex) :
  (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger p).matrix
        other post =
      QuantumBlockEncoding.Coeff.rat 0 ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
              p ↑source).daggerCleanup.proved =
        false ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
              p ↑source).unitaryExtension.proved =
        false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract_daggerOffCandidate_zero
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source other post :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits :
    QuantumBlockEncoding.clog2 p.kappa =
      3)
  (hsource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑source =
      true)
  (hotherSource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑other =
      true)
  (hpost :
    ↑post =
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
          p ↑source).postSwapImageIndex)
  (hne :
    ↑other ≠
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
          p
          ↑source).candidatePreimageIndex) :
  (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
            p).matrix
        other post =
      QuantumBlockEncoding.Coeff.rat 0 ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
              p
              ↑source).daggerCleanup.proved =
        false ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
              p
              ↑source).unitaryExtension.proved =
        false
    Off-candidate dagger entries are zero on the active global-slot source domain.

For the fixed post-SWAP target in the global-slot inverse-on-range contract,
any other active global-source preimage whose index is not the named candidate
cannot have a transpose-style `(O_D^BS)^†` entry into that target.  This is a
matrix-entry bridge only: it does not promote dagger cleanup or unitary flags.
Theorem10.2.381
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access global slot inverse on range contract restricted dagger column cleanup”; the hypotheses and conclusion in the code panel fix its exact scope. Restricted active-domain dagger-column cleanup for the global-slot route.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Restricted active-domain dagger-column cleanup for the global-slot route. For the contract post-SWAP target, the named candidate has dagger entry '1', and every other active global-source row has dagger entry '0'. This is only a column statement over 'bandedSparseAccessPaperGlobalSlotSource'; it does not promote semantic cleanup, unitarity, circuit-unitarity, or block-extraction obligations.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:7883. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.381●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract_restrictedDaggerColumnCleanup
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits : QuantumBlockEncoding.clog2 p.kappa = 3)
  (hsource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑source =
      true) :
  ∃ post pre,
    QuantumBlockEncoding.GHL2025.BandedSparseAccessPostSwapCleanup p
        source post pre ∧
      ↑post =
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
              p ↑source).postSwapImageIndex ∧
        ↑pre =
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                p ↑source).candidatePreimageIndex ∧
          QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
                p ↑pre =
              true ∧
            (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
                      p).matrix
                  pre post =
                QuantumBlockEncoding.Coeff.rat 1 ∧
              (∀
                  (other :
                    Fin
                      (QuantumBlockEncoding.qubitDim
                        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
                          p))),
                  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
                        p ↑other =
                      true →
                    ↑other ≠ ↑pre →
                      (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
                              p).matrix
                          other post =
                        QuantumBlockEncoding.Coeff.rat 0) ∧
                (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                          p ↑source).daggerCleanup.proved =
                    false ∧
                  (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                            p ↑source).unitaryExtension.proved =
                      false ∧
                    (QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract
                              p).daggerCleanup.proved =
                        false ∧
                      (QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract
                                p).unitaryExtension.proved =
                          false ∧
                        (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
                                  p).unitary.proved =
                            false ∧
                          (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
                                    p).unitary.proved =
                              false ∧
                            (QuantumBlockEncoding.GHL2025.defaultOneTermRobinTheoremData
                                        p).obligations.circuitUnitary.proved =
                                false ∧
                              (QuantumBlockEncoding.GHL2025.defaultOneTermRobinTheoremData
                                        p).obligations.blockExtraction.proved =
                                false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract_restrictedDaggerColumnCleanup
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits :
    QuantumBlockEncoding.clog2 p.kappa =
      3)
  (hsource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑source =
      true) :
  ∃ post pre,
    QuantumBlockEncoding.GHL2025.BandedSparseAccessPostSwapCleanup
        p source post pre ∧
      ↑post =
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
              p
              ↑source).postSwapImageIndex ∧
        ↑pre =
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                p
                ↑source).candidatePreimageIndex ∧
          QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
                p ↑pre =
              true ∧
            (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
                      p).matrix
                  pre post =
                QuantumBlockEncoding.Coeff.rat
                  1 ∧
              (∀
                  (other :
                    Fin
                      (QuantumBlockEncoding.qubitDim
                        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
                          p))),
                  QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
                        p ↑other =
                      true →
                    ↑other ≠ ↑pre →
                      (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
                              p).matrix
                          other post =
                        QuantumBlockEncoding.Coeff.rat
                          0) ∧
                (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                          p
                          ↑source).daggerCleanup.proved =
                    false ∧
                  (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                            p
                            ↑source).unitaryExtension.proved =
                      false ∧
                    (QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract
                              p).daggerCleanup.proved =
                        false ∧
                      (QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract
                                p).unitaryExtension.proved =
                          false ∧
                        (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
                                  p).unitary.proved =
                            false ∧
                          (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
                                    p).unitary.proved =
                              false ∧
                            (QuantumBlockEncoding.GHL2025.defaultOneTermRobinTheoremData
                                        p).obligations.circuitUnitary.proved =
                                false ∧
                              (QuantumBlockEncoding.GHL2025.defaultOneTermRobinTheoremData
                                        p).obligations.blockExtraction.proved =
                                false
    Restricted active-domain dagger-column cleanup for the global-slot route.

For the contract post-SWAP target, the named candidate has dagger entry `1`,
and every other active global-source row has dagger entry `0`.  This is only a
column statement over `bandedSparseAccessPaperGlobalSlotSource`; it does not
promote semantic cleanup, unitarity, circuit-unitarity, or block-extraction
obligations.
Theorem10.2.382
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access global slot inverse on range contract restricted dagger column indicator”; the hypotheses and conclusion in the code panel fix its exact scope. Indicator form of the restricted active-domain dagger column.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Indicator form of the restricted active-domain dagger column. This is the same active-source-only column statement as 'bandedSparseAccessGlobalSlotInverseOnRangeContract_restrictedDaggerColumnCleanup', rewritten as one if-then-else formula. It does not promote inverse, cleanup, unitarity, circuit-unitarity, or block-extraction obligations.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:7972. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.382●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract_restrictedDaggerColumnIndicator
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits : QuantumBlockEncoding.clog2 p.kappa = 3)
  (hsource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑source =
      true) :
  ∃ post pre,
    QuantumBlockEncoding.GHL2025.BandedSparseAccessPostSwapCleanup p
        source post pre ∧
      ↑post =
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
              p ↑source).postSwapImageIndex ∧
        ↑pre =
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                p ↑source).candidatePreimageIndex ∧
          QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
                p ↑pre =
              true ∧
            (∀
                (other :
                  Fin
                    (QuantumBlockEncoding.qubitDim
                      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
                        p))),
                QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
                      p ↑other =
                    true →
                  (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
                          p).matrix
                      other post =
                    if ↑other = ↑pre then
                      QuantumBlockEncoding.Coeff.rat 1
                    else QuantumBlockEncoding.Coeff.rat 0) ∧
              (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                        p ↑source).inverseOnRange.proved =
                  false ∧
                (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                          p ↑source).uniquePreimage.proved =
                    false ∧
                  (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                            p
                            ↑source).imageInjectiveOnGlobalSource.proved =
                      false ∧
                    (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                              p ↑source).daggerCleanup.proved =
                        false ∧
                      (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                                p ↑source).unitaryExtension.proved =
                          false ∧
                        (QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract
                                  p).daggerCleanup.proved =
                            false ∧
                          (QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract
                                    p).unitaryExtension.proved =
                              false ∧
                            (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
                                      p).unitary.proved =
                                false ∧
                              (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
                                        p).unitary.proved =
                                  false ∧
                                (QuantumBlockEncoding.GHL2025.defaultOneTermRobinTheoremData
                                            p).obligations.circuitUnitary.proved =
                                    false ∧
                                  (QuantumBlockEncoding.GHL2025.defaultOneTermRobinTheoremData
                                            p).obligations.blockExtraction.proved =
                                    false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract_restrictedDaggerColumnIndicator
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (source :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (hn : 3 ≤ p.n) (hkappa : p.kappa = 7)
  (hκbits :
    QuantumBlockEncoding.clog2 p.kappa =
      3)
  (hsource :
    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
        p ↑source =
      true) :
  ∃ post pre,
    QuantumBlockEncoding.GHL2025.BandedSparseAccessPostSwapCleanup
        p source post pre ∧
      ↑post =
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
              p
              ↑source).postSwapImageIndex ∧
        ↑pre =
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                p
                ↑source).candidatePreimageIndex ∧
          QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
                p ↑pre =
              true ∧
            (∀
                (other :
                  Fin
                    (QuantumBlockEncoding.qubitDim
                      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
                        p))),
                QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
                      p ↑other =
                    true →
                  (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
                          p).matrix
                      other post =
                    if ↑other = ↑pre then
                      QuantumBlockEncoding.Coeff.rat
                        1
                    else
                      QuantumBlockEncoding.Coeff.rat
                        0) ∧
              (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                        p
                        ↑source).inverseOnRange.proved =
                  false ∧
                (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                          p
                          ↑source).uniquePreimage.proved =
                    false ∧
                  (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                            p
                            ↑source).imageInjectiveOnGlobalSource.proved =
                      false ∧
                    (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                              p
                              ↑source).daggerCleanup.proved =
                        false ∧
                      (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                                p
                                ↑source).unitaryExtension.proved =
                          false ∧
                        (QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract
                                  p).daggerCleanup.proved =
                            false ∧
                          (QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract
                                    p).unitaryExtension.proved =
                              false ∧
                            (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
                                      p).unitary.proved =
                                false ∧
                              (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
                                        p).unitary.proved =
                                  false ∧
                                (QuantumBlockEncoding.GHL2025.defaultOneTermRobinTheoremData
                                            p).obligations.circuitUnitary.proved =
                                    false ∧
                                  (QuantumBlockEncoding.GHL2025.defaultOneTermRobinTheoremData
                                            p).obligations.blockExtraction.proved =
                                    false
    Indicator form of the restricted active-domain dagger column.

This is the same active-source-only column statement as
`bandedSparseAccessGlobalSlotInverseOnRangeContract_restrictedDaggerColumnCleanup`,
rewritten as one if-then-else formula.  It does not promote inverse,
cleanup, unitarity, circuit-unitarity, or block-extraction obligations.
Definition10.2.383
uses 0used by 0✓L∃∀N

Plain-English reading. This type lists the allowed alternatives for “banded sparse access cleanup scope”; its constructors are the cases that downstream code must handle. Allowed scopes for the next 'O_D^BS' cleanup theorem packet.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Allowed scopes for the next 'O_D^BS' cleanup theorem packet. This is planning data for Phase 1 faithful-paper work. Selecting a scope here does not prove cleanup, full-domain injectivity, or unitary extension.

Declaration kind. inductive.

Source: QuantumBlockEncoding/GHL2025.lean:8059. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.383●1 definition
  • inductive(3 constructors)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    inductive QuantumBlockEncoding.GHL2025.BandedSparseAccessCleanupScope : Type
    inductive QuantumBlockEncoding.GHL2025.BandedSparseAccessCleanupScope :
  Type
    Allowed scopes for the next `O_D^BS` cleanup theorem packet.

This is planning data for Phase 1 faithful-paper work.  Selecting a scope here
does not prove cleanup, full-domain injectivity, or unitary extension.

    Constructors

    activeGlobalSource :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessCleanupScope
    fullCleanDomain :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessCleanupScope
    fullSpace :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessCleanupScope
Definition10.2.384
uses 0used by 0✓L∃∀N

Plain-English reading. This record groups the data and proof fields needed for “banded sparse access cleanup scope decision”. A proposition-valued field is a requirement until a constructor supplies it. Non-promoting decision for the next 'O_D^BS' cleanup theorem domain.

Formal status. Data contract in the default import surface; proposition-valued fields are obligations, not automatically established facts.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Non-promoting decision for the next 'O_D^BS' cleanup theorem domain. The current compiled matrix-entry theorem is restricted to active global-source rows. Full clean-domain cleanup still needs a reversible image rule for every clean unused sparse branch, and full-space cleanup/unitarity still needs a separate reversible-extension argument. This record keeps that boundary machine-checkable before any lower proof packet tries to promote 'daggerCleanup'.

Declaration kind. structure.

Source: QuantumBlockEncoding/GHL2025.lean:8075. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.384●1 definition
  • structure(10 fields)defined in QuantumBlockEncoding/GHL2025.lean
    complete
    structure QuantumBlockEncoding.GHL2025.BandedSparseAccessCleanupScopeDecision :
  Type
    structure QuantumBlockEncoding.GHL2025.BandedSparseAccessCleanupScopeDecision :
  Type
    Non-promoting decision for the next `O_D^BS` cleanup theorem domain.

The current compiled matrix-entry theorem is restricted to active global-source
rows.  Full clean-domain cleanup still needs a reversible image rule for every
clean unused sparse branch, and full-space cleanup/unitarity still needs a
separate reversible-extension argument.  This record keeps that boundary
machine-checkable before any lower proof packet tries to promote
`daggerCleanup`.

    Fields

    sourceAnchor : String
    selectedScope : QuantumBlockEncoding.GHL2025.BandedSparseAccessCleanupScope
    selectedPredicate : String
    selectedEvidence : String
    fullCleanDomainSelected : Bool
    fullSpaceSelected : Bool
    semanticCleanupPromotionAllowed : Bool
    paperContractCleanup : QuantumBlockEncoding.GHL2025.ObligationRecord
    fullCleanDomainCleanup : QuantumBlockEncoding.GHL2025.ObligationRecord
    fullSpaceUnitaryExtension : QuantumBlockEncoding.GHL2025.ObligationRecord
Definition10.2.385
uses 0used by 0✓L∃∀N

Plain-English reading. This definition gives the library's named construction or computation for “banded sparse access cleanup scope decision”. Default cleanup-scope decision after the restricted dagger-column indicator.

Formal status. Compiled declaration in the default ASPBE import surface; its kind and displayed Lean type determine how it may be used.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Default cleanup-scope decision after the restricted dagger-column indicator. The selected theorem domain is active global-source only. The surrounding obligation records are copied from the existing paper and full-domain contracts so their 'proved = false' status stays synchronized with the actual contracts.

Declaration kind. def.

Source: QuantumBlockEncoding/GHL2025.lean:8095. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Definition10.2.385●1 definition
  • defdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessCleanupScopeDecision
    def QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GHL2025.BandedSparseAccessCleanupScopeDecision
    Default cleanup-scope decision after the restricted dagger-column indicator.

The selected theorem domain is active global-source only.  The surrounding
obligation records are copied from the existing paper and full-domain contracts
so their `proved = false` status stays synchronized with the actual contracts.
Theorem10.2.386
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access cleanup scope decision active global source”; the hypotheses and conclusion in the code panel fix its exact scope. The cleanup-scope decision selects the active global-source theorem and keeps all broader cleanup/unitarity obligations closed to proof-flag promotion.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The cleanup-scope decision selects the active global-source theorem and keeps all broader cleanup/unitarity obligations closed to proof-flag promotion.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:8116. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.386●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision_activeGlobalSource
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
          p).selectedScope =
      QuantumBlockEncoding.GHL2025.BandedSparseAccessCleanupScope.activeGlobalSource ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
            p).selectedPredicate =
        "bandedSparseAccessPaperGlobalSlotSource" ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
              p).selectedEvidence =
          "bandedSparseAccessGlobalSlotInverseOnRangeContract_restrictedDaggerColumnIndicator" ∧
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
                p).fullCleanDomainSelected =
            false ∧
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
                  p).fullSpaceSelected =
              false ∧
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
                    p).semanticCleanupPromotionAllowed =
                false ∧
              (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
                        p).paperContractCleanup.proved =
                  false ∧
                (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
                          p).fullCleanDomainCleanup.proved =
                    false ∧
                  (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
                          p).fullSpaceUnitaryExtension.proved =
                    false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision_activeGlobalSource
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
          p).selectedScope =
      QuantumBlockEncoding.GHL2025.BandedSparseAccessCleanupScope.activeGlobalSource ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
            p).selectedPredicate =
        "bandedSparseAccessPaperGlobalSlotSource" ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
              p).selectedEvidence =
          "bandedSparseAccessGlobalSlotInverseOnRangeContract_restrictedDaggerColumnIndicator" ∧
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
                p).fullCleanDomainSelected =
            false ∧
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
                  p).fullSpaceSelected =
              false ∧
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
                    p).semanticCleanupPromotionAllowed =
                false ∧
              (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
                        p).paperContractCleanup.proved =
                  false ∧
                (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
                          p).fullCleanDomainCleanup.proved =
                    false ∧
                  (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
                          p).fullSpaceUnitaryExtension.proved =
                    false
    The cleanup-scope decision selects the active global-source theorem and keeps
all broader cleanup/unitarity obligations closed to proof-flag promotion.
Theorem10.2.387
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access cleanup scope decision prior pde source transcript guard”; the hypotheses and conclusion in the code panel fix its exact scope. The cleanup-scope decision does not accept the prior PDE sparse-access transcript as a full-space unitary-extension proof.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The cleanup-scope decision does not accept the prior PDE sparse-access transcript as a full-space unitary-extension proof. This is a guard for the next Phase 1 source-contract packet: the prior paper's equation is recorded as a source anchor, while its resource proof and any Robin-specific reversible-extension use remain unproved in QBE.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:8147. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.387●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision_priorPDESourceTranscriptGuard
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
          p).selectedScope =
      QuantumBlockEncoding.GHL2025.BandedSparseAccessCleanupScope.activeGlobalSource ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
            p).fullSpaceSelected =
        false ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
                p).fullSpaceUnitaryExtension.proved =
          false ∧
        QuantumBlockEncoding.GHL2025.bandedSparseAccessPriorPDESourceContract.oracleEquation =
            "O_A^BS |0>^(n-l)|s>^l|i>^n = |r_si>^n|i>^n" ∧
          QuantumBlockEncoding.GHL2025.bandedSparseAccessPriorPDESourceContract.resourceClaim.proved =
              false ∧
            QuantumBlockEncoding.GHL2025.bandedSparseAccessPriorPDESourceContract.robinUnusedBranchImageRule =
                none ∧
              QuantumBlockEncoding.GHL2025.bandedSparseAccessPriorPDESourceContract.closesUnusedZeroBranchExtension =
                  false ∧
                QuantumBlockEncoding.GHL2025.bandedSparseAccessPriorPDESourceContract.lowerProofSearchAllowed =
                    false ∧
                  (QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract
                          p).unitaryExtension.proved =
                    false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision_priorPDESourceTranscriptGuard
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
          p).selectedScope =
      QuantumBlockEncoding.GHL2025.BandedSparseAccessCleanupScope.activeGlobalSource ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
            p).fullSpaceSelected =
        false ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
                p).fullSpaceUnitaryExtension.proved =
          false ∧
        QuantumBlockEncoding.GHL2025.bandedSparseAccessPriorPDESourceContract.oracleEquation =
            "O_A^BS |0>^(n-l)|s>^l|i>^n = |r_si>^n|i>^n" ∧
          QuantumBlockEncoding.GHL2025.bandedSparseAccessPriorPDESourceContract.resourceClaim.proved =
              false ∧
            QuantumBlockEncoding.GHL2025.bandedSparseAccessPriorPDESourceContract.robinUnusedBranchImageRule =
                none ∧
              QuantumBlockEncoding.GHL2025.bandedSparseAccessPriorPDESourceContract.closesUnusedZeroBranchExtension =
                  false ∧
                QuantumBlockEncoding.GHL2025.bandedSparseAccessPriorPDESourceContract.lowerProofSearchAllowed =
                    false ∧
                  (QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract
                          p).unitaryExtension.proved =
                    false
    The cleanup-scope decision does not accept the prior PDE sparse-access
transcript as a full-space unitary-extension proof.

This is a guard for the next Phase 1 source-contract packet: the prior paper's
equation is recorded as a source anchor, while its resource proof and any
Robin-specific reversible-extension use remain unproved in QBE.
Theorem10.2.388
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “banded sparse access cleanup scope decision full clean domain image rule blocked”; the hypotheses and conclusion in the code panel fix its exact scope. The cleanup-scope decision keeps the full clean-domain image-rule slot blocked.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. The cleanup-scope decision keeps the full clean-domain image-rule slot blocked. This guard ties the active-global-source scope choice to the unused-branch source decision and the full clean-domain wrapper. It is not a cleanup or unitarity theorem: the missing image rule remains 'none', proof search remains disabled, and every full clean-domain semantic field stays false.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:8177. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.388●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision_fullCleanDomainImageRuleBlocked
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
          p).selectedScope =
      QuantumBlockEncoding.GHL2025.BandedSparseAccessCleanupScope.activeGlobalSource ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
            p).fullCleanDomainSelected =
        false ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
              p).semanticCleanupPromotionAllowed =
          false ∧
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
                  p).fullCleanDomainCleanup.proved =
            false ∧
          QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision.lowerProofSearchAllowed =
              false ∧
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
                    p j).proposedImageIndex =
                none ∧
              ((QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                          p).unusedBranchImageRuleContract
                      j).proposedImageIndex =
                  none ∧
                ((QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                              p).unusedBranchImageRuleContract
                          j).imageSpecified.proved =
                    false ∧
                  (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                            p).unusedBranchImageSpecified.proved =
                      false ∧
                    (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                              p).fullCleanDomainInjective.proved =
                        false ∧
                      (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                                p).daggerCleanup.proved =
                          false ∧
                        (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                                p).unitaryExtension.proved =
                          false
    theorem QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision_fullCleanDomainImageRuleBlocked
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
          p).selectedScope =
      QuantumBlockEncoding.GHL2025.BandedSparseAccessCleanupScope.activeGlobalSource ∧
    (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
            p).fullCleanDomainSelected =
        false ∧
      (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
              p).semanticCleanupPromotionAllowed =
          false ∧
        (QuantumBlockEncoding.GHL2025.bandedSparseAccessCleanupScopeDecision
                  p).fullCleanDomainCleanup.proved =
            false ∧
          QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedZeroBranchSourceDecision.lowerProofSearchAllowed =
              false ∧
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessUnusedBranchImageRuleContract
                    p
                    j).proposedImageIndex =
                none ∧
              ((QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                          p).unusedBranchImageRuleContract
                      j).proposedImageIndex =
                  none ∧
                ((QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                              p).unusedBranchImageRuleContract
                          j).imageSpecified.proved =
                    false ∧
                  (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                            p).unusedBranchImageSpecified.proved =
                      false ∧
                    (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                              p).fullCleanDomainInjective.proved =
                        false ∧
                      (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                                p).daggerCleanup.proved =
                          false ∧
                        (QuantumBlockEncoding.GHL2025.bandedSparseAccessFullCleanDomainExtensionContract
                                p).unitaryExtension.proved =
                          false
    The cleanup-scope decision keeps the full clean-domain image-rule slot blocked.

This guard ties the active-global-source scope choice to the unused-branch
source decision and the full clean-domain wrapper.  It is not a cleanup or
unitarity theorem: the missing image rule remains `none`, proof search remains
disabled, and every full clean-domain semantic field stays false.
Theorem10.2.389
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “default banded sparse access paper contract cleanup route bridge boundary column n 3”; the hypotheses and conclusion in the code panel fix its exact scope. Concrete boundary-source regression for the default paper-contract cleanup route.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Concrete boundary-source regression for the default paper-contract cleanup route. The historical source column '48' is outside the rejected row-dependent valid source predicate, but it is an active global-slot source for Lemma 1. This instance routes that column through the default cleanup bridge without promoting any 'O_D^BS' semantic proof flag.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:8216. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.389●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract_cleanupRouteBridge_boundaryColumn_n3 :
  let p :=
    { n := 3, kappa := 7, functionPieces := 1,
      polynomialDegreeCost := 1 };
  have source := ⟨48, ⋯⟩;
  ∃ post pre,
    QuantumBlockEncoding.GHL2025.BandedSparseAccessPostSwapCleanup p
        source post pre ∧
      ↑post =
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
              p ↑source).postSwapImageIndex ∧
        ↑pre =
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                p ↑source).candidatePreimageIndex ∧
          QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
                p ↑source =
              true ∧
            QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource
                  p ↑source =
                false ∧
              QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
                    p ↑pre =
                  true ∧
                (∀
                    (pre' :
                      Fin
                        (QuantumBlockEncoding.qubitDim
                          (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
                            p))),
                    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
                          p ↑pre' =
                        true →
                      QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
                            p ↑pre' =
                          ↑post →
                        ↑pre' = ↑pre) ∧
                  (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
                            p).matrix
                        pre post =
                      QuantumBlockEncoding.Coeff.rat 1 ∧
                    (QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract
                              p).daggerCleanup.proved =
                        false ∧
                      (QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract
                                p).unitaryExtension.proved =
                          false ∧
                        (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
                                  p).unitary.proved =
                            false ∧
                          (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
                                  p).unitary.proved =
                            false
    theorem QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract_cleanupRouteBridge_boundaryColumn_n3 :
  let p :=
    { n := 3, kappa := 7,
      functionPieces := 1,
      polynomialDegreeCost := 1 };
  have source := ⟨48, ⋯⟩;
  ∃ post pre,
    QuantumBlockEncoding.GHL2025.BandedSparseAccessPostSwapCleanup
        p source post pre ∧
      ↑post =
          (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
              p
              ↑source).postSwapImageIndex ∧
        ↑pre =
            (QuantumBlockEncoding.GHL2025.bandedSparseAccessGlobalSlotInverseOnRangeContract
                p
                ↑source).candidatePreimageIndex ∧
          QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
                p ↑source =
              true ∧
            QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperValidCleanSource
                  p ↑source =
                false ∧
              QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
                    p ↑pre =
                  true ∧
                (∀
                    (pre' :
                      Fin
                        (QuantumBlockEncoding.qubitDim
                          (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
                            p))),
                    QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperGlobalSlotSource
                          p ↑pre' =
                        true →
                      QuantumBlockEncoding.GHL2025.bandedSparseAccessPaperImage
                            p ↑pre' =
                          ↑post →
                        ↑pre' = ↑pre) ∧
                  (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
                            p).matrix
                        pre post =
                      QuantumBlockEncoding.Coeff.rat
                        1 ∧
                    (QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract
                              p).daggerCleanup.proved =
                        false ∧
                      (QuantumBlockEncoding.GHL2025.defaultBandedSparseAccessPaperContract
                                p).unitaryExtension.proved =
                          false ∧
                        (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS
                                  p).unitary.proved =
                            false ∧
                          (QuantumBlockEncoding.GHL2025.oneTermRobinGate_O_D_BS_dagger
                                  p).unitary.proved =
                            false
    Concrete boundary-source regression for the default paper-contract cleanup route.

The historical source column `48` is outside the rejected row-dependent valid
source predicate, but it is an active global-slot source for Lemma 1.  This
instance routes that column through the default cleanup bridge without
promoting any `O_D^BS` semantic proof flag.
Theorem10.2.390
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “robin indicator bit position ge”; the hypotheses and conclusion in the code panel fix its exact scope. Cycle 12: robinIndicatorBitPosition = 1 + 2*p.n, hence >= 1 + p.n.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Cycle 12: robinIndicatorBitPosition = 1 + 2*p.n, hence >= 1 + p.n.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:8265. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.390●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.robinIndicatorBitPosition_ge
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GHL2025.robinIndicatorBitPosition p ≥ 1 + p.n
    theorem QuantumBlockEncoding.GHL2025.robinIndicatorBitPosition_ge
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GHL2025.robinIndicatorBitPosition
      p ≥
    1 + p.n
    Cycle 12: robinIndicatorBitPosition = 1 + 2*p.n, hence >= 1 + p.n.
Theorem10.2.391
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “indicator oracle image system val preserved”; the hypotheses and conclusion in the code panel fix its exact scope. Cycle 12: The system register value is preserved by indicatorOracleImage.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Cycle 12: The system register value is preserved by indicatorOracleImage. XORing with a bit at position indPos = 1 + 2n does not affect bits [1, 1+n). main.tex:1088-1099 -

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:8275. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.391●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleImage_systemVal_preserved
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  QuantumBlockEncoding.GHL2025.indicatorOracleImage p j >>> 1 &&&
      1 <<< p.n - 1 =
    j >>> 1 &&& 1 <<< p.n - 1
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleImage_systemVal_preserved
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  QuantumBlockEncoding.GHL2025.indicatorOracleImage
          p j >>>
        1 &&&
      1 <<< p.n - 1 =
    j >>> 1 &&& 1 <<< p.n - 1
    Cycle 12: The system register value is preserved by indicatorOracleImage.
XORing with a bit at position indPos = 1 + 2n does not affect bits [1, 1+n).
main.tex:1088-1099 -
Theorem10.2.392
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “indicator oracle image is bulk preserved”; the hypotheses and conclusion in the code panel fix its exact scope. Cycle 12: The isBulk predicate gives the same result for j and indicatorOracleImage p j, because isBulk only depends on the system register value, which is preserved.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Cycle 12: The isBulk predicate gives the same result for j and indicatorOracleImage p j, because isBulk only depends on the system register value, which is preserved. main.tex:1088-1099 -

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:8289. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.392●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleImage_isBulk_preserved
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  (if
        2 ≤
            QuantumBlockEncoding.GHL2025.indicatorOracleImage p j >>>
                1 &&&
              1 <<< p.n - 1 ∧
          QuantumBlockEncoding.GHL2025.indicatorOracleImage p j >>>
                1 &&&
              1 <<< p.n - 1 ≤
            QuantumBlockEncoding.gridSize p.n - 3 then
      1
    else 0) =
    if
        2 ≤ j >>> 1 &&& 1 <<< p.n - 1 ∧
          j >>> 1 &&& 1 <<< p.n - 1 ≤
            QuantumBlockEncoding.gridSize p.n - 3 then
      1
    else 0
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleImage_isBulk_preserved
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  (if
        2 ≤
            QuantumBlockEncoding.GHL2025.indicatorOracleImage
                  p j >>>
                1 &&&
              1 <<< p.n - 1 ∧
          QuantumBlockEncoding.GHL2025.indicatorOracleImage
                  p j >>>
                1 &&&
              1 <<< p.n - 1 ≤
            QuantumBlockEncoding.gridSize
                p.n -
              3 then
      1
    else 0) =
    if
        2 ≤ j >>> 1 &&& 1 <<< p.n - 1 ∧
          j >>> 1 &&& 1 <<< p.n - 1 ≤
            QuantumBlockEncoding.gridSize
                p.n -
              3 then
      1
    else 0
    Cycle 12: The isBulk predicate gives the same result for j and indicatorOracleImage p j,
because isBulk only depends on the system register value, which is preserved.
main.tex:1088-1099 -
Theorem10.2.393
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “indicator oracle image self inverse”; the hypotheses and conclusion in the code panel fix its exact scope. Cycle 12: General self-inverse property for indicatorOracleImage.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Cycle 12: General self-inverse property for indicatorOracleImage. Applying the indicator oracle image twice returns the original value for all j, because the indicator bit is XORed twice (and isBulk is preserved). main.tex:1088-1099 -

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:8304. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.393●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleImage_self_inverse
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) (j : ℕ) :
  QuantumBlockEncoding.GHL2025.indicatorOracleImage p
      (QuantumBlockEncoding.GHL2025.indicatorOracleImage p j) =
    j
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleImage_self_inverse
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j : ℕ) :
  QuantumBlockEncoding.GHL2025.indicatorOracleImage
      p
      (QuantumBlockEncoding.GHL2025.indicatorOracleImage
        p j) =
    j
    Cycle 12: General self-inverse property for indicatorOracleImage.
Applying the indicator oracle image twice returns the original value for all j,
because the indicator bit is XORed twice (and isBulk is preserved).
main.tex:1088-1099 -
Theorem10.2.394
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “one term robin gate u indic dagger self inverse bridge”; the hypotheses and conclusion in the code panel fix its exact scope. Source-facing bridge for the explicit 'U_indic^dagger' transcript slot.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Source-facing bridge for the explicit 'U_indic^dagger' transcript slot. The dagger slot uses the same matrix as 'U_indic' because the underlying indicator image is self-inverse. This theorem records the bridge used by the conversion window; it does not change the active backend gate list.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:8319. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.394●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGate_U_indic_dagger_selfInverseBridge
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  (QuantumBlockEncoding.GHL2025.oneTermRobinGate_U_indic_dagger
          p).matrix =
      (QuantumBlockEncoding.GHL2025.oneTermRobinGate_U_indic p).matrix ∧
    ∀ (j : ℕ),
      QuantumBlockEncoding.GHL2025.indicatorOracleImage p
          (QuantumBlockEncoding.GHL2025.indicatorOracleImage p j) =
        j
    theorem QuantumBlockEncoding.GHL2025.oneTermRobinGate_U_indic_dagger_selfInverseBridge
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  (QuantumBlockEncoding.GHL2025.oneTermRobinGate_U_indic_dagger
          p).matrix =
      (QuantumBlockEncoding.GHL2025.oneTermRobinGate_U_indic
          p).matrix ∧
    ∀ (j : ℕ),
      QuantumBlockEncoding.GHL2025.indicatorOracleImage
          p
          (QuantumBlockEncoding.GHL2025.indicatorOracleImage
            p j) =
        j
    Source-facing bridge for the explicit `U_indic^dagger` transcript slot.

The dagger slot uses the same matrix as `U_indic` because the underlying
indicator image is self-inverse.  This theorem records the bridge used by the
conversion window; it does not change the active backend gate list.
Theorem10.2.395
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “indicator oracle image injective”; the hypotheses and conclusion in the code panel fix its exact scope. Cycle 12: General injectivity for indicatorOracleImage, derived from self-inverse.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Cycle 12: General injectivity for indicatorOracleImage, derived from self-inverse. main.tex:1088-1099 -

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:8330. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.395●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleImage_injective
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) {j₁ j₂ : ℕ}
  (h :
    QuantumBlockEncoding.GHL2025.indicatorOracleImage p j₁ =
      QuantumBlockEncoding.GHL2025.indicatorOracleImage p j₂) :
  j₁ = j₂
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleImage_injective
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  {j₁ j₂ : ℕ}
  (h :
    QuantumBlockEncoding.GHL2025.indicatorOracleImage
        p j₁ =
      QuantumBlockEncoding.GHL2025.indicatorOracleImage
        p j₂) :
  j₁ = j₂
    Cycle 12: General injectivity for indicatorOracleImage, derived from self-inverse.
main.tex:1088-1099 -
Theorem10.2.396
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “robin indicator bit position lt total qubits”; the hypotheses and conclusion in the code panel fix its exact scope. Cycle 12: robinIndicatorBitPosition is strictly below oneTermRobinTotalQubits.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Cycle 12: robinIndicatorBitPosition is strictly below oneTermRobinTotalQubits. indPos = 1 + 2n < 2n + clog2 n + clog2 fp + 5 = totalQubits, since clog2 ≥ 0 and 5 > 1.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:8341. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.396●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.robinIndicatorBitPosition_lt_totalQubits
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GHL2025.robinIndicatorBitPosition p <
    QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p
    theorem QuantumBlockEncoding.GHL2025.robinIndicatorBitPosition_lt_totalQubits
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  QuantumBlockEncoding.GHL2025.robinIndicatorBitPosition
      p <
    QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
      p
    Cycle 12: robinIndicatorBitPosition is strictly below oneTermRobinTotalQubits.
indPos = 1 + 2n < 2n + clog2 n + clog2 fp + 5 = totalQubits, since clog2 ≥ 0 and 5 > 1.
Theorem10.2.397
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “indicator oracle image lt”; the hypotheses and conclusion in the code panel fix its exact scope. Cycle 12: indicatorOracleImage preserves the qubitDim bound.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Cycle 12: indicatorOracleImage preserves the qubitDim bound. When j < 2^totalQubits, the image is also < 2^totalQubits, because the XOR operand is either 0 or a single bit at position indPos < totalQubits.

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:8352. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.397●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleImage_lt
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) {j : ℕ}
  (hj :
    j <
      QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)) :
  QuantumBlockEncoding.GHL2025.indicatorOracleImage p j <
    QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleImage_lt
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  {j : ℕ}
  (hj :
    j <
      QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)) :
  QuantumBlockEncoding.GHL2025.indicatorOracleImage
      p j <
    QuantumBlockEncoding.qubitDim
      (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
        p)
    Cycle 12: indicatorOracleImage preserves the qubitDim bound.
When j < 2^totalQubits, the image is also < 2^totalQubits, because the XOR
operand is either 0 or a single bit at position indPos < totalQubits.
Theorem10.2.398
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “indicator oracle image bijective”; the hypotheses and conclusion in the code panel fix its exact scope. Cycle 12: Bijectivity of indicatorOracleImage on the Fin domain.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Cycle 12: Bijectivity of indicatorOracleImage on the Fin domain. A self-inverse function on a finite type is bijective: injective by cancellation, surjective because image(image(j)) = j. main.tex:1088-1099 -

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:8370. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.398●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleImage_bijective
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  (∀
      (a b :
        Fin
          (QuantumBlockEncoding.qubitDim
            (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))),
      ⟨QuantumBlockEncoding.GHL2025.indicatorOracleImage p ↑a, ⋯⟩ =
          ⟨QuantumBlockEncoding.GHL2025.indicatorOracleImage p ↑b, ⋯⟩ →
        a = b) ∧
    ∀
      (y :
        Fin
          (QuantumBlockEncoding.qubitDim
            (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))),
      ∃ x,
        ⟨QuantumBlockEncoding.GHL2025.indicatorOracleImage p ↑x, ⋯⟩ = y
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleImage_bijective
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  (∀
      (a b :
        Fin
          (QuantumBlockEncoding.qubitDim
            (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
              p))),
      ⟨QuantumBlockEncoding.GHL2025.indicatorOracleImage
              p ↑a,
            ⋯⟩ =
          ⟨QuantumBlockEncoding.GHL2025.indicatorOracleImage
              p ↑b,
            ⋯⟩ →
        a = b) ∧
    ∀
      (y :
        Fin
          (QuantumBlockEncoding.qubitDim
            (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
              p))),
      ∃ x,
        ⟨QuantumBlockEncoding.GHL2025.indicatorOracleImage
              p ↑x,
            ⋯⟩ =
          y
    Cycle 12: Bijectivity of indicatorOracleImage on the Fin domain.
A self-inverse function on a finite type is bijective:
injective by cancellation, surjective because image(image(j)) = j.
main.tex:1088-1099 -
Theorem10.2.399
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “indicator oracle matrix col has one”; the hypotheses and conclusion in the code panel fix its exact scope. Cycle 12: For each column j, there is exactly one row i with M[i][j] = 1, namely i = ⟨indicatorOracleImage p j.val, ...⟩.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Cycle 12: For each column j, there is exactly one row i with M[i][j] = 1, namely i = ⟨indicatorOracleImage p j.val, ...⟩. main.tex:1088-1099 -

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:8393. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.399●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleMatrix_col_has_one
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))) :
  QuantumBlockEncoding.GHL2025.indicatorOracleMatrix p
      ⟨QuantumBlockEncoding.GHL2025.indicatorOracleImage p ↑j, ⋯⟩ j =
    QuantumBlockEncoding.Coeff.rat 1
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleMatrix_col_has_one
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p))) :
  QuantumBlockEncoding.GHL2025.indicatorOracleMatrix
      p
      ⟨QuantumBlockEncoding.GHL2025.indicatorOracleImage
          p ↑j,
        ⋯⟩
      j =
    QuantumBlockEncoding.Coeff.rat 1
    Cycle 12: For each column j, there is exactly one row i with M[i][j] = 1,
namely i = ⟨indicatorOracleImage p j.val, ...⟩.
main.tex:1088-1099 -
Theorem10.2.400
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “indicator oracle matrix col unique”; the hypotheses and conclusion in the code panel fix its exact scope. Cycle 12: For each column j, any row i with M[i][j] = 1 must equal ⟨indicatorOracleImage p j.val, ...⟩, so the 1-entry is unique per column.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Cycle 12: For each column j, any row i with M[i][j] = 1 must equal ⟨indicatorOracleImage p j.val, ...⟩, so the 1-entry is unique per column. main.tex:1088-1099 -

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:8405. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.400●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleMatrix_col_unique
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (i j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (h :
    QuantumBlockEncoding.GHL2025.indicatorOracleMatrix p i j =
      QuantumBlockEncoding.Coeff.rat 1) :
  i = ⟨QuantumBlockEncoding.GHL2025.indicatorOracleImage p ↑j, ⋯⟩
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleMatrix_col_unique
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (i j :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (h :
    QuantumBlockEncoding.GHL2025.indicatorOracleMatrix
        p i j =
      QuantumBlockEncoding.Coeff.rat 1) :
  i =
    ⟨QuantumBlockEncoding.GHL2025.indicatorOracleImage
        p ↑j,
      ⋯⟩
    Cycle 12: For each column j, any row i with M[i][j] = 1 must equal
⟨indicatorOracleImage p j.val, ...⟩, so the 1-entry is unique per column.
main.tex:1088-1099 -
Theorem10.2.401
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “indicator oracle matrix row has one”; the hypotheses and conclusion in the code panel fix its exact scope. Cycle 12: For each row i, there exists a column j with M[i][j] = 1, from surjectivity of indicatorOracleImage.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Cycle 12: For each row i, there exists a column j with M[i][j] = 1, from surjectivity of indicatorOracleImage. main.tex:1088-1099 -

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:8418. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.401●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleMatrix_row_has_one
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (i :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))) :
  ∃ j,
    QuantumBlockEncoding.GHL2025.indicatorOracleMatrix p i j =
      QuantumBlockEncoding.Coeff.rat 1
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleMatrix_row_has_one
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (i :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p))) :
  ∃ j,
    QuantumBlockEncoding.GHL2025.indicatorOracleMatrix
        p i j =
      QuantumBlockEncoding.Coeff.rat 1
    Cycle 12: For each row i, there exists a column j with M[i][j] = 1,
from surjectivity of indicatorOracleImage.
main.tex:1088-1099 -
Theorem10.2.402
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “indicator oracle matrix row unique”; the hypotheses and conclusion in the code panel fix its exact scope. Cycle 12: For each row i, the column j with M[i][j] = 1 is unique, from injectivity of indicatorOracleImage.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Cycle 12: For each row i, the column j with M[i][j] = 1 is unique, from injectivity of indicatorOracleImage. main.tex:1088-1099 -

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:8436. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.402●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleMatrix_row_unique
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (i j₁ j₂ :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p)))
  (h₁ :
    QuantumBlockEncoding.GHL2025.indicatorOracleMatrix p i j₁ =
      QuantumBlockEncoding.Coeff.rat 1)
  (h₂ :
    QuantumBlockEncoding.GHL2025.indicatorOracleMatrix p i j₂ =
      QuantumBlockEncoding.Coeff.rat 1) :
  j₁ = j₂
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleMatrix_row_unique
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters)
  (i j₁ j₂ :
    Fin
      (QuantumBlockEncoding.qubitDim
        (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
          p)))
  (h₁ :
    QuantumBlockEncoding.GHL2025.indicatorOracleMatrix
        p i j₁ =
      QuantumBlockEncoding.Coeff.rat 1)
  (h₂ :
    QuantumBlockEncoding.GHL2025.indicatorOracleMatrix
        p i j₂ =
      QuantumBlockEncoding.Coeff.rat 1) :
  j₁ = j₂
    Cycle 12: For each row i, the column j with M[i][j] = 1 is unique,
from injectivity of indicatorOracleImage.
main.tex:1088-1099 -
Theorem10.2.403
uses 0used by 0✓L∃∀N

Plain-English reading. Lean checks the proposition indexed as “indicator oracle matrix is permutation”; the hypotheses and conclusion in the code panel fix its exact scope. Cycle 12: indicatorOracleMatrix is a permutation matrix: each row has exactly one entry equal to 1, and each column has exactly one entry equal to 1.

Formal status. Compiled theorem in the default ASPBE import surface; the displayed Lean signature is the authoritative claim.

Why it is in this chapter. Paper-facing backend models and concrete Robin-boundary example artifacts.

Technical source note. Cycle 12: indicatorOracleMatrix is a permutation matrix: each row has exactly one entry equal to 1, and each column has exactly one entry equal to 1. This follows from indicatorOracleImage being a bijection on the Fin domain. main.tex:1088-1099 -

Declaration kind. theorem.

Source: QuantumBlockEncoding/GHL2025.lean:8452. A commit-pinned external link is added by the publication build when the source exists at the published ref.

Lean code for Theorem10.2.403●1 theorem
  • theoremdefined in QuantumBlockEncoding/GHL2025.lean
    complete
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleMatrix_is_permutation
  (p : QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  (∀
      (i :
        Fin
          (QuantumBlockEncoding.qubitDim
            (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))),
      ∃ j,
        QuantumBlockEncoding.GHL2025.indicatorOracleMatrix p i j =
            QuantumBlockEncoding.Coeff.rat 1 ∧
          ∀
            (j' :
              Fin
                (QuantumBlockEncoding.qubitDim
                  (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
                    p))),
            QuantumBlockEncoding.GHL2025.indicatorOracleMatrix p i j' =
                QuantumBlockEncoding.Coeff.rat 1 →
              j' = j) ∧
    ∀
      (j :
        Fin
          (QuantumBlockEncoding.qubitDim
            (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits p))),
      ∃ i,
        QuantumBlockEncoding.GHL2025.indicatorOracleMatrix p i j =
            QuantumBlockEncoding.Coeff.rat 1 ∧
          ∀
            (i' :
              Fin
                (QuantumBlockEncoding.qubitDim
                  (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
                    p))),
            QuantumBlockEncoding.GHL2025.indicatorOracleMatrix p i' j =
                QuantumBlockEncoding.Coeff.rat 1 →
              i' = i
    theorem QuantumBlockEncoding.GHL2025.indicatorOracleMatrix_is_permutation
  (p :
    QuantumBlockEncoding.GHL2025.OneTermRobinParameters) :
  (∀
      (i :
        Fin
          (QuantumBlockEncoding.qubitDim
            (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
              p))),
      ∃ j,
        QuantumBlockEncoding.GHL2025.indicatorOracleMatrix
              p i j =
            QuantumBlockEncoding.Coeff.rat
              1 ∧
          ∀
            (j' :
              Fin
                (QuantumBlockEncoding.qubitDim
                  (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
                    p))),
            QuantumBlockEncoding.GHL2025.indicatorOracleMatrix
                  p i j' =
                QuantumBlockEncoding.Coeff.rat
                  1 →
              j' = j) ∧
    ∀
      (j :
        Fin
          (QuantumBlockEncoding.qubitDim
            (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
              p))),
      ∃ i,
        QuantumBlockEncoding.GHL2025.indicatorOracleMatrix
              p i j =
            QuantumBlockEncoding.Coeff.rat
              1 ∧
          ∀
            (i' :
              Fin
                (QuantumBlockEncoding.qubitDim
                  (QuantumBlockEncoding.GHL2025.oneTermRobinTotalQubits
                    p))),
            QuantumBlockEncoding.GHL2025.indicatorOracleMatrix
                  p i' j =
                QuantumBlockEncoding.Coeff.rat
                  1 →
              i' = i
    Cycle 12: indicatorOracleMatrix is a permutation matrix: each row has exactly
one entry equal to 1, and each column has exactly one entry equal to 1.
This follows from indicatorOracleImage being a bijection on the Fin domain.
main.tex:1088-1099 -