AutoSamplingTheory.SALD.cycle51DiscreteForwardKlSkeletonUpperPacket

Data definition / provenance and workflow record

Meaning and type

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

def cycle51DiscreteForwardKlSkeletonUpperPacket :
    DiscreteForwardKlUpperPacket

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.

objective : String (explicit)

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

Cycle 51 upper: consume the cycle-49 analytic readiness audit and the cycle-50 continuous derivative/DV handoff, then re-wire thm:forward-KL-discrete through the explicit EM endpoint/conditional-Fokker-Planck, frozen-defect/LSI, DV velocity, Gronwall, and accumulated-error interfaces matching main_body.tex:299-323 and appendix.tex:260-592.

sourceLabels : List String (explicit)

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

Ordered data items

1. thm:forward-KL-discrete
2. proof:thm:forward-KL-discrete
3. eq:frozen_interp_terminal_disc_prop_additive_final
4. proof:thm:forward-KL-discrete:conditional-fp
5. proof:thm:forward-KL-discrete:derivative
6. proof:thm:forward-KL-discrete:dv-velocity
7. proof:thm:forward-KL-discrete:gronwall
8. proof:thm:forward-KL-discrete:accumulated-error
9. thm:forward-KL
10. proof:thm:forward-KL:derivative
11. eq:LSI-KL-FI
12. lem:dv_variation
13. lem:gronwall

modeDiscipline : List String (explicit)

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

Ordered data items

1. faithfulPaper Phase 1: use only original main_body.tex:299-323 and appendix.tex:260-592 for the discrete theorem; sald_version_2.tex remains excluded.
2. Before lower work, the five slow backends are explicitly checked: SALD.saldGronwallEndpointCalculusContract for endpoint-safe Gronwall, dvVariationalFormulaInterface saldDvVariationSource plus SALD.saldDvFiniteLogMgfContract for DV, SALD.saldLsiKlFiDensityTestContract for LSI/KL/FI, SALD.forwardKlDerivativeCandidateContract plus cycle-50 scalar handoff for continuous derivative reuse, and SALD.discreteForwardKlEmInterpolationSideConditionContract for EM endpoint/conditional-law Fokker-Planck.
3. The discrete route remains EM interpolation -> conditional-FP/KL derivative -> frozen score defect and LSI -> DV velocity bound -> s-to-t time change -> Gronwall -> linear-slowdown accumulated-error collection.
4. The theorem constants T/(r*alpha), 2*r*eta^2*barGamma/alpha', (1/r)*A_alpha(pi,v), and 2*r*eta*barDelta_{alpha'} are preserved exactly.

nonGoals : List String (explicit)

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

Ordered data items

1. Do not restate thm:forward-KL-discrete or promote SALD.discreteSaldContract above contractOnly.
2. Do not prove or replace the EM conditional-Fokker-Planck backend, frozen-defect lemma, LSI/KL/FI density-test, source-cited DV formula, Gronwall, endpoint stitching, or accumulated-error bridge in this upper packet.
3. Do not add density, absolute-continuity, endpoint, finite-log-mgf, coefficient-regularity, or stitched-interval assumptions to the theorem statement.
4. Do not start broad SLT or measure-theory backfill before the discrete theorem-level route and lower packet are synchronized.

lowerPacket : List String (explicit)

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

Ordered data items

1. Middle should synchronize this cycle-51 discrete route with conversion-windows/ASTIS-SALD-001.md and proof-obligations/ASTIS-SALD-001.md, preserving two-way Lean/Markdown/source mapping.
2. Lower should target exactly SALD.discreteForwardKlDerivativeCandidateContract / SALD.discreteForwardKlDerivativeObligation / sald.discrete_forward_kl.kl_derivative over appendix.tex:334-491.
3. First lower sub-slice: use the existing source-cited EM endpoint and conditional-Fokker-Planck interfaces from appendix.tex:334-385 as inputs to the KL derivative identity; do not attempt to prove the conditional-FP theorem from scratch in this cycle.
4. Second lower sub-slice: preserve the frozen-cross coefficient, the moving Young share, eq:LSI-KL-FI, and the DV velocity witness through appendix.tex:454-523 before any Gronwall display work.
5. If a measure-theory fact is missing, refine the named EM, density, absolute-continuity, or stitched-interval obligation instead of changing the theorem statement.

reviewerChecklist : List String (explicit)

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

Ordered data items

1. SALD.discreteSaldContract lists SALD.cycle51DiscreteForwardKlSkeletonObligation while remaining contractOnly.
2. SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle51_theorem_interface_route after the cycle-46 route audit and before lower EM/defect/DV/Gronwall nodes.
3. SALD.saldDependenciesForLabel "thm:forward-KL-discrete" includes the cycle-51 upper packet, obligation, DAG node, and the cycle-50 derivative/DV lower handoff as a reusable coefficient dependency.
4. All five slow analytic backends remain ProofStatus.obligation or ProofStatus.sourceCited unless their full analytic dependencies build locally.
5. python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass.

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 cycle51DiscreteForwardKlSkeletonUpperPacket :
    DiscreteForwardKlUpperPacket where
  objective := "Cycle 51 upper: consume the cycle-49 analytic readiness audit and the cycle-50 continuous derivative/DV handoff, then re-wire thm:forward-KL-discrete through the explicit EM endpoint/conditional-Fokker-Planck, frozen-defect/LSI, DV velocity, Gronwall, and accumulated-error interfaces matching main_body.tex:299-323 and appendix.tex:260-592."
  sourceLabels := [
    "thm:forward-KL-discrete",
    "proof:thm:forward-KL-discrete",
    "eq:frozen_interp_terminal_disc_prop_additive_final",
    "proof:thm:forward-KL-discrete:conditional-fp",
    "proof:thm:forward-KL-discrete:derivative",
    "proof:thm:forward-KL-discrete:dv-velocity",
    "proof:thm:forward-KL-discrete:gronwall",
    "proof:thm:forward-KL-discrete:accumulated-error",
    "thm:forward-KL",
    "proof:thm:forward-KL:derivative",
    "eq:LSI-KL-FI",
    "lem:dv_variation",
    "lem:gronwall"
  ]
  modeDiscipline := [
    "faithfulPaper Phase 1: use only original main_body.tex:299-323 and appendix.tex:260-592 for the discrete theorem; sald_version_2.tex remains excluded.",
    "Before lower work, the five slow backends are explicitly checked: SALD.saldGronwallEndpointCalculusContract for endpoint-safe Gronwall, dvVariationalFormulaInterface saldDvVariationSource plus SALD.saldDvFiniteLogMgfContract for DV, SALD.saldLsiKlFiDensityTestContract for LSI/KL/FI, SALD.forwardKlDerivativeCandidateContract plus cycle-50 scalar handoff for continuous derivative reuse, and SALD.discreteForwardKlEmInterpolationSideConditionContract for EM endpoint/conditional-law Fokker-Planck.",
    "The discrete route remains EM interpolation -> conditional-FP/KL derivative -> frozen score defect and LSI -> DV velocity bound -> s-to-t time change -> Gronwall -> linear-slowdown accumulated-error collection.",
    "The theorem constants T/(r*alpha), 2*r*eta^2*barGamma/alpha', (1/r)*A_alpha(pi,v), and 2*r*eta*barDelta_{alpha'} are preserved exactly."
  ]
  nonGoals := [
    "Do not restate thm:forward-KL-discrete or promote SALD.discreteSaldContract above contractOnly.",
    "Do not prove or replace the EM conditional-Fokker-Planck backend, frozen-defect lemma, LSI/KL/FI density-test, source-cited DV formula, Gronwall, endpoint stitching, or accumulated-error bridge in this upper packet.",
    "Do not add density, absolute-continuity, endpoint, finite-log-mgf, coefficient-regularity, or stitched-interval assumptions to the theorem statement.",
    "Do not start broad SLT or measure-theory backfill before the discrete theorem-level route and lower packet are synchronized."
  ]
  lowerPacket := [
    "Middle should synchronize this cycle-51 discrete route with conversion-windows/ASTIS-SALD-001.md and proof-obligations/ASTIS-SALD-001.md, preserving two-way Lean/Markdown/source mapping.",
    "Lower should target exactly SALD.discreteForwardKlDerivativeCandidateContract / SALD.discreteForwardKlDerivativeObligation / sald.discrete_forward_kl.kl_derivative over appendix.tex:334-491.",
    "First lower sub-slice: use the existing source-cited EM endpoint and conditional-Fokker-Planck interfaces from appendix.tex:334-385 as inputs to the KL derivative identity; do not attempt to prove the conditional-FP theorem from scratch in this cycle.",
    "Second lower sub-slice: preserve the frozen-cross coefficient, the moving Young share, eq:LSI-KL-FI, and the DV velocity witness through appendix.tex:454-523 before any Gronwall display work.",
    "If a measure-theory fact is missing, refine the named EM, density, absolute-continuity, or stitched-interval obligation instead of changing the theorem statement."
  ]
  reviewerChecklist := [
    "SALD.discreteSaldContract lists SALD.cycle51DiscreteForwardKlSkeletonObligation while remaining contractOnly.",
    "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle51_theorem_interface_route after the cycle-46 route audit and before lower EM/defect/DV/Gronwall nodes.",
    "SALD.saldDependenciesForLabel \"thm:forward-KL-discrete\" includes the cycle-51 upper packet, obligation, DAG node, and the cycle-50 derivative/DV lower handoff as a reusable coefficient dependency.",
    "All five slow analytic backends remain ProofStatus.obligation or ProofStatus.sourceCited unless their full analytic dependencies build locally.",
    "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass."
  ]
  status := ProofStatus.obligation

/-- Cycle-51 obligation tying the post-cycle-50 discrete route back to the
source-cited EM/Fokker--Planck interfaces. -/