Exercise PLAN-502

Exact Hermitian clean blocks admit guaranteed adjoint-paired Chebyshev transformations: E^k gives T_{2k}(A) and UR E^k gives T_{2k+1}(A) after projection. U and U† have directed between-plane maps; neither distinct planes nor failure of every special unpaired construction follows from U≠U†.

1. For an interior eigenvalue, define unit flag-orthogonal companions q=(U†Jv-tJv)/sqrt(1-t^2), q'=(UJv-tJv)/sqrt(1-t^2) and their input/output planes, not assumed geometrically distinct.
2. Unitarity gives directed U:H→H' and U†:H'→H with the same symmetric matrix [[t,s],[s,-t]], rather than a bidirectional-U swap assertion.
3. The flag reflection is diag(1,-1) in both ordered companion bases.
4. Thus E=U†RUR has matrix SZSZ=M(t)^2 and rotates the input plane by twice arccos(t).
5. For k≥1, E^k has clean coefficient T_{2k}(t) and flag-orthogonal second-kind coefficient sqrt(1-t^2)U_{2k-1}(t); k=0 and t=±1 are treated directly.
6. The directed extra UR yields clean coefficient T_{2k+1}(t); spectral assembly proves the two projected operator identities, not complete clean-output-state equalities.
7. UR already has degree-one clean block A. A rotation completion shows higher unpaired failure even with H=H', while the explicit non-Hermitian U_* with clean block diag(1/2,1) has correct unpaired clean blocks at every power; pairing is a general guarantee, not universal necessity.

Write p=Jv and s=sqrt(1-t^2) for a unit eigenvector v with |t|<1. The exact clean compression is Hermitian, so J†U†J=A as well. Define q=(U†p-tp)/s and q'=(Up-tp)/s. Both are unit vectors orthogonal to the entire clean subspace: their clean components vanish by the compression identities, and their squared leakage norms are1-t^2 by unitarity. Let B=(p,q), B'=(p,q') and H=span(B), H'=span(B'). Then Up=tp+s q', U†p=tp+s q, and applying U to the latter identity gives Uq=s p-t q'. Similarly U†q'=s p-t q. Thus U:H→H' and U†:H'→H have the same matrix S=[[t,s],[s,-t]] between these directed ordered bases. This statement does not imply that the two geometric planes differ or that U itself maps H' back to H.

The flag reflection is Z=diag(1,-1) in both bases. Therefore [E]_B=SZSZ=M(t)^2, with M(t)=[[t,-s],[s,t]]. If theta=arccos(t)∈(0,pi), then s=sin(theta) and M(t) is the rotation by theta. Hence E^k on H has matrix M(t)^(2k), with diagonal T_{2k}(t). For k≥1 its upper-right entry is -s U_{2k-1}(t), where U_l denotes the second-kind Chebyshev polynomial, not the bare encoding unitary U. In particular E^k p=T_{2k}(t)p+s U_{2k-1}(t)q; k=0 is the identity directly. The second term is generally nonzero and flag-orthogonal.

UR maps the ordered input basis B into the output basis B' with matrix SZ=M(t). Consequently UR E^k has between-basis matrix M(t)^(2k+1), and UR E^k p=T_{2k+1}(t)p+s U_{2k}(t)q'. Its clean coefficient is T_{2k+1}(t), even though its full image can have leakage into q'. This is an odd between-plane map, not an assertion that UR alone can be iterated using one fixed matrix in one fixed ordered basis. An orthonormal eigenbasis of A spans the input clean subspace, so projecting and using linearity gives J†E^kJ=T_{2k}(A) and J†UR E^kJ=T_{2k+1}(A).

If t=+1 or-1, unitarity forces Up=tp and U†p=tp, since the clean component already has norm1. The clean line is invariant, E p=p, and UR E^k p=t p. These agree with T_{2k}(±1)=1 and T_{2k+1}(±1)=±1. No companion is obtained by dividing zero leakage by zero, and no genuinely two-dimensional sector is required at these endpoints. The compression is a contraction (||Ax||≤||UJx||=||x||), so all Hermitian eigenvalues are in[-1,1]; this is a consequence, not an added input promise.

The unpaired conclusion is a boundary of the guarantee, not universal failure. Since RJ=J, J†URJ=A=T_1(A) for every admissible U. For a failure at higher order, take t=1/2, s=sqrt(3)/2, A=t I_2 and U=[[t,-s],[s,t]] tensor I_2, with the ancilla first. This real rotation completion is unitary and non-Hermitian and has clean block A. But UR=[[t,s],[s,-t]] tensor I_2 is a reflection, so (UR)^2=I_4 and its clean block is I_2, whereas T_2(A)=(-1/2)I_2. For any system eigenvector v in this same example, q=-|1>v and q'=|1>v, so H=H' as geometric subspaces even though the ordered companion bases differ. Thus a different-plane necessity is false, and coincidence of the planes alone does not give the desired unpaired powers.

Conversely non-Hermiticity need not spoil any unpaired clean polynomial block. In ancilla-major order |00>,|01>,|10>,|11>, take U_*= [[t,0,s,0],[0,1,0,0],[s,0,-t,0],[0,0,0,i]], A_*=diag(t,1), t=1/2, s=sqrt(3)/2. This U_* is unitary but not Hermitian because of the last i phase; its clean block is the Hermitian A_*. On span{|00>,|10>}, U_*R is the rotation [[t,-s],[s,t]], while |01> has eigenvalue1 and |11> has eigenvalue-i. Therefore its clean block at every integer power j≥0 is diag(T_j(t),1)=T_j(A_*). This all-j conclusion follows from the invariant-block calculation, not a finite test. The unused endpoint leakage direction allows the global non-Hermitian phase. It refutes any claim that U≠U† alone universally prevents unpaired Chebyshev blocks.

The source-paired formulas are sufficient guarantees for every exact normalized completion, whether or not special unpaired constructions also work. They identify compressions of unitary operators, not deterministic normalized state outputs. If a projected operator annihilates an input, there is no normalized accepted output for that input; otherwise normalization after a successful projection must still be applied. No gate-compilation, finite-bit or preparation/readout cost claim is made, and a differently scaled or approximate block encoding would require the corresponding normalized target and error analysis.

1. Constructs normalized flag-orthogonal companions for |t|<1 and the directed matrices U:H→H' and U†:H'→H, without assuming H and H' differ or that U is its own reverse. (1 weight)
2. Uses R=diag(1,-1) in both ordered bases to prove E=SZSZ=M(t)^2 and the twice-arccos(t) rotation on H. (1 weight)
3. Derives the projected T_{2k}(A) and T_{2k+1}(A) identities, identifies the second-kind leakage structure where used, and handles k=0 and t=±1 without zero-norm division or a whole-state equality. (1 weight)
4. Distinguishes a guaranteed paired construction from universal necessity: UR has clean block A; valid counterexamples show failure of a higher unpaired power, possible H=H', and a non-Hermitian completion with correct unpaired clean blocks at all orders. (1 weight)

No aligned Lean view has been admitted for this native core in this textbook checkpoint. This is not a claim that the underlying mathematics lacks a formal proof. Contributions must preserve this exact target contract and split.

Read the Lean track and existing diagnostic warmups

• website/research/sources.json

  lin-wiebe-2026-qasc

  Source SHA-256: 9f85f45f20ed5a46931ad0dec3b94ee09748bc56e39fae29b0aefe910440b0ae · MIT metadata only
• Primary source

Contract: 98b38dad470c122798e05fd2a04d0eb423be4db3152139647aba7ea1f6d7c040

Item digest: e27a2d656b19db77c0bd30e1d3dd79991f217b19010ecb9ef70e6835e4578349

Graph snapshot: qcb-snapshot-ffa644f0da3031a1ce754b9a

Authored support steps: 7. This is not measured graph distance or model difficulty.

{ "question_version": 2, "predecessor_key": "qcb-core-plan-502-parity-pairing-even-odd-iterate", "predecessor_item_digest": "6dc47a40ade49740a2dfd6231b609dda5a32ae284283b606e270b133f68ad1f5", "predecessor_contract_hash": "ca69caa5a76a137ff31eb17382680dd340e5907b0417a2939624c317c195e037", "current_contract_hash": "98b38dad470c122798e05fd2a04d0eb423be4db3152139647aba7ea1f6d7c040", "changed_contract_fields": [ "question" ], "repair_record": "shared/repairs/plan-502-paired-iterate-question-successor.json", "repair_record_digest": "e315e85057c5ec86dccbf36f86132679cd2921e9c09b0ebb7c40e46f896bf56c", "review_record": "reports/textbook-v1/repairs/plan-502-paired-iterate-question-successor-review.json", "review_digest": "3c747f35af07d3006db21c0a8dc5e89a07be546d22034541e46084693283609e", "equivalence_to_predecessor_claimed": false, "additional_core_count": 0 }