AutoSamplingTheory.SALD.cycle16UnifiedForwardKlTransportBridgeMiddleContract

Data definition / provenance and workflow record

Meaning and type

The result has data type AutoSamplingTheory.SALD.UnifiedForwardKlTransportBridgeMiddleContract. A value of this type stores descriptions; it is not a proof of the statements in those descriptions.

def cycle16UnifiedForwardKlTransportBridgeMiddleContract :
    UnifiedForwardKlTransportBridgeMiddleContract

Construction and field-by-field explanation

Construct a data record from explicit fields and the audited defaults shown below.

This Lean definition constructs provenance or workflow data. It does not prove the mathematical statements stored as text. Status labels, named dependencies and citations are data, not compilation, proof or source certificates.

residualIdentitySource : AutoSamplingTheory.SourceAnchor (explicit)

Nested provenance data.

AutoSamplingTheory.SALD.saldGuidedResidualMainSource — audited data reference, not expanded and not a compiled dependency edge

correctionEquationSource : AutoSamplingTheory.SourceAnchor (explicit)

Nested provenance data.

AutoSamplingTheory.SALD.saldCorrectionFieldSource — audited data reference, not expanded and not a compiled dependency edge

transportBridgeSource : AutoSamplingTheory.SourceAnchor (explicit)

Nested provenance data.

AutoSamplingTheory.SALD.saldUnifiedTransportBridgeSource — audited data reference, not expanded and not a compiled dependency edge

specializationProofSource : AutoSamplingTheory.SourceAnchor (explicit)

Nested provenance data.

AutoSamplingTheory.SALD.saldUnifiedForwardKlProofSource — audited data reference, not expanded and not a compiled dependency edge

objective : String (explicit)

Descriptive text even when field names say formula, theorem, inequality, derivative, source gap, or proof.

Translate the cycle-16 upper packet into a lower-ready line ledger for the unified VA-SALD transport bridge, without changing the theorem statement or proving the general theorem.

sourceAlgebraLedger : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. main_body.tex:359-363 states partial_t pi_t+div(pi_t*u_t)=-pi_t*(g_t-E_pi_t[g_t]) and points back to prop:guided_path_residual.
2. main_body.tex:364-367 introduces w_t through div(pi_t*w_t)=pi_t*(g_t-E_pi_t[g_t]).
3. main_body.tex:368 concludes by cancellation that u_t+w_t is a transport velocity field for pi_t.
4. appendix.tex:949-951 proves thm:unified-forward-KL only by specializing thm:general-moving-target-SALD with c_t<-u_t.

leanInterfaceMap : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. SALD.guidedResidualIdentityContract and sald.guided_path_residual.identity provide the centered residual identity.
2. SALD.unifiedForwardKlSpecializationContract records the correction-field equation, v_t=u_t+w_t, c_t=u_t, and m_t=w_t.
3. SALD.cycle16UnifiedForwardKlTransportBridgeLowerContract isolates the signed cancellation and divergence-linearity backend for the lower slice.
4. SALD.unifiedForwardKlTransportBridgeObligation is the lower target for the local divergence cancellation.
5. SALD.unifiedForwardKlSpecializationObligation remains the theorem-level specialization after the transport bridge is available.

citedResultInterfaces : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. No SLT theorem applies to this bridge; it is local continuity-equation and divergence algebra.
2. The later residual DV, Gronwall, and discrete EM uses stay under their existing obligations.

obligations : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. sald.guided_path_residual.identity
2. sald.unified_forward_kl.transport_bridge_lower
3. sald.unified_forward_kl.transport_velocity_bridge
4. sald.unified_forward_kl.specialization

lowerPacket : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. Target sald.unified_forward_kl.transport_velocity_bridge only.
2. Prove or refine only the cancellation between the two displayed equations in main_body.tex:359-368.
3. Use SALD.cycle16UnifiedForwardKlTransportBridgeLowerContract for the exact residual equation, correction equation, divergence-linearity, and cancellation ledger.
4. Keep existence, regularity, boundary conditions, and weak/divergence interpretation for w_t as explicit local obligations.
5. After the bridge, record v_t=u_t+w_t, c_t=u_t, and m_t=w_t; do not start DV, Gronwall, or discrete EM proof search.

reviewerChecklist : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. The line ledger cites main_body.tex:359-368 and appendix.tex:949-951, not sald_version_2.tex.
2. SALD.unifiedForwardKlTransportBridgeObligation remains an obligation and is not replaced by a direct VA-SALD proof.
3. SALD.unifiedForwardKlSpecializationObligation still depends on sald.unified_forward_kl.transport_velocity_bridge.
4. No correction-field existence theorem or SLT reuse claim is marked formalized.

status : AutoSamplingTheory.ProofStatus (explicit)

Stored workflow tag; honor the exact default but do not infer mathematical certification.

AutoSamplingTheory.ProofStatus.obligation — stored label only; no proof certification

def cycle16UnifiedForwardKlTransportBridgeMiddleContract :
    UnifiedForwardKlTransportBridgeMiddleContract where
  residualIdentitySource := saldGuidedResidualMainSource
  correctionEquationSource := saldCorrectionFieldSource
  transportBridgeSource := saldUnifiedTransportBridgeSource
  specializationProofSource := saldUnifiedForwardKlProofSource
  objective := "Translate the cycle-16 upper packet into a lower-ready line ledger for the unified VA-SALD transport bridge, without changing the theorem statement or proving the general theorem."
  sourceAlgebraLedger := [
    "main_body.tex:359-363 states partial_t pi_t+div(pi_t*u_t)=-pi_t*(g_t-E_pi_t[g_t]) and points back to prop:guided_path_residual.",
    "main_body.tex:364-367 introduces w_t through div(pi_t*w_t)=pi_t*(g_t-E_pi_t[g_t]).",
    "main_body.tex:368 concludes by cancellation that u_t+w_t is a transport velocity field for pi_t.",
    "appendix.tex:949-951 proves thm:unified-forward-KL only by specializing thm:general-moving-target-SALD with c_t<-u_t."
  ]
  leanInterfaceMap := [
    "SALD.guidedResidualIdentityContract and sald.guided_path_residual.identity provide the centered residual identity.",
    "SALD.unifiedForwardKlSpecializationContract records the correction-field equation, v_t=u_t+w_t, c_t=u_t, and m_t=w_t.",
    "SALD.cycle16UnifiedForwardKlTransportBridgeLowerContract isolates the signed cancellation and divergence-linearity backend for the lower slice.",
    "SALD.unifiedForwardKlTransportBridgeObligation is the lower target for the local divergence cancellation.",
    "SALD.unifiedForwardKlSpecializationObligation remains the theorem-level specialization after the transport bridge is available."
  ]
  citedResultInterfaces := [
    "No SLT theorem applies to this bridge; it is local continuity-equation and divergence algebra.",
    "The later residual DV, Gronwall, and discrete EM uses stay under their existing obligations."
  ]
  obligations := [
    "sald.guided_path_residual.identity",
    "sald.unified_forward_kl.transport_bridge_lower",
    "sald.unified_forward_kl.transport_velocity_bridge",
    "sald.unified_forward_kl.specialization"
  ]
  lowerPacket := [
    "Target sald.unified_forward_kl.transport_velocity_bridge only.",
    "Prove or refine only the cancellation between the two displayed equations in main_body.tex:359-368.",
    "Use SALD.cycle16UnifiedForwardKlTransportBridgeLowerContract for the exact residual equation, correction equation, divergence-linearity, and cancellation ledger.",
    "Keep existence, regularity, boundary conditions, and weak/divergence interpretation for w_t as explicit local obligations.",
    "After the bridge, record v_t=u_t+w_t, c_t=u_t, and m_t=w_t; do not start DV, Gronwall, or discrete EM proof search."
  ]
  reviewerChecklist := [
    "The line ledger cites main_body.tex:359-368 and appendix.tex:949-951, not sald_version_2.tex.",
    "SALD.unifiedForwardKlTransportBridgeObligation remains an obligation and is not replaced by a direct VA-SALD proof.",
    "SALD.unifiedForwardKlSpecializationObligation still depends on sald.unified_forward_kl.transport_velocity_bridge.",
    "No correction-field existence theorem or SLT reuse claim is marked formalized."
  ]
  status := ProofStatus.obligation