AutoSamplingTheory.SALD.cycle56DiscreteForwardKlSkeletonUpperPacket

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 cycle56DiscreteForwardKlSkeletonUpperPacket :
    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 56 upper: wire thm:forward-KL-discrete through the existing source-cited EM endpoint/conditional-Fokker-Planck interfaces, cycle-51 derivative/LSI scalar handoff, DV velocity witness, and appendix.tex:526-592 Gronwall/accumulated-error bridge, matching main_body.tex:299-323 without changing constants or theorem status.

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. eq:LSI-KL-FI
11. lem:dv_variation
12. lem:gronwall

modeDiscipline : List String (explicit)

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

Ordered data items

1. faithfulPaper Phase 1: keep main_body.tex:299-323 and appendix.tex:260-592 fixed; sald_version_2.tex remains excluded.
2. Five-backend check 1, Gronwall: SALD.saldGronwallEndpointCalculusContract and SALD.discreteForwardKlGronwallInstantiationContract expose endpoint-safe differentiability/FTC, stitched intervals, coefficient regularity, and exponent rewrite obligations.
3. Five-backend check 2, DV: dvVariationalFormulaInterface saldDvVariationSource and SALD.discreteForwardKlDvFiniteLogMgfWitnessContract expose common-space, absolute-continuity, finite KL, finite log-mgf, measurability, and positive-alpha scaling for nu=hat rho_s, mu=tilde pi_s.
4. Five-backend check 3, LSI/KL/FI: SALD.saldLsiKlFiDensityTestContract and probability.lsi_to_kl_fi expose density, zero-set convention, admissible sqrt-density test, entropy identity, finite KL/FI, and Fisher chain-rule obligations.
5. Five-backend check 4, continuous derivative: SALD.forwardKlDerivativeCandidateContract, SALD.forwardKlDerivativeSideConditionContract, and the cycle-55 mass handoff keep the continuous Fokker-Planck/KL derivative backend visible for reused transport and schedule side conditions without promoting it.
6. Five-backend check 5, EM interpolation: SALD.discreteForwardKlEmInterpolationSideConditionContract exposes endpoint laws, conditional drift, conditional-law density, conditional Fokker-Planck, Laplacian split, and stitched interval interfaces as source-cited obligations.

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, change SALD.discreteForwardKlStatementContract, 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, residual exponent, or accumulated-error bridge in this upper packet.
3. Do not add hidden density, absolute-continuity, endpoint, finite-log-mgf, boundary, coefficient-regularity, inverse-schedule, or stitched-interval assumptions to the theorem statement.
4. Do not start broad SLT, concentration, measure-theory, or SDE backfill until this theorem-level route remains stable under review.

lowerPacket : List String (explicit)

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

Ordered data items

1. Middle should synchronize this cycle-56 upper route with conversion-windows/ASTIS-SALD-001.md and proof-obligations/ASTIS-SALD-001.md, preserving the two-way Lean/Markdown/source map.
2. Lower should target exactly SALD.discreteForwardKlGronwallInstantiationContract / SALD.discreteForwardKlGronwallAccumulationObligation / sald.discrete_forward_kl.gronwall_accumulation over appendix.tex:526-592.
3. First lower sub-slice: consume the existing sald.discrete_forward_kl.kl_derivative and sald.discrete_forward_kl.dv_velocity_bound interfaces to instantiate the source a(t) and b(t) in Eq. KL-derivative-7-discrete.
4. Second lower sub-slice: keep endpoint law stitching and K(T)=KL(rho_K^eta||pi_T), K(0)=KL(rho_0||pi_0) as named side conditions rather than new theorem assumptions.
5. Third lower sub-slice: route the linear slowdown t(s)=s/r, residual exponent, barGamma, and barDelta collection to SALD.discreteForwardKlAccumulatedErrorBridgeContract without changing the main-body coefficients.

reviewerChecklist : List String (explicit)

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

Ordered data items

1. SALD.discreteSaldContract lists SALD.cycle56DiscreteForwardKlSkeletonObligation while remaining contractOnly.
2. SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle56_theorem_interface_route after the cycle-51 derivative lower route and before the reusable EM/defect/DV/Gronwall nodes.
3. SALD.saldDependenciesForLabel "thm:forward-KL-discrete" includes the cycle-56 upper packet, obligation, DAG node, and the explicit EM/Fokker-Planck interface names.
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 cycle56DiscreteForwardKlSkeletonUpperPacket :
    DiscreteForwardKlUpperPacket where
  objective := "Cycle 56 upper: wire thm:forward-KL-discrete through the existing source-cited EM endpoint/conditional-Fokker-Planck interfaces, cycle-51 derivative/LSI scalar handoff, DV velocity witness, and appendix.tex:526-592 Gronwall/accumulated-error bridge, matching main_body.tex:299-323 without changing constants or theorem status."
  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",
    "eq:LSI-KL-FI",
    "lem:dv_variation",
    "lem:gronwall"
  ]
  modeDiscipline := [
    "faithfulPaper Phase 1: keep main_body.tex:299-323 and appendix.tex:260-592 fixed; sald_version_2.tex remains excluded.",
    "Five-backend check 1, Gronwall: SALD.saldGronwallEndpointCalculusContract and SALD.discreteForwardKlGronwallInstantiationContract expose endpoint-safe differentiability/FTC, stitched intervals, coefficient regularity, and exponent rewrite obligations.",
    "Five-backend check 2, DV: dvVariationalFormulaInterface saldDvVariationSource and SALD.discreteForwardKlDvFiniteLogMgfWitnessContract expose common-space, absolute-continuity, finite KL, finite log-mgf, measurability, and positive-alpha scaling for nu=hat rho_s, mu=tilde pi_s.",
    "Five-backend check 3, LSI/KL/FI: SALD.saldLsiKlFiDensityTestContract and probability.lsi_to_kl_fi expose density, zero-set convention, admissible sqrt-density test, entropy identity, finite KL/FI, and Fisher chain-rule obligations.",
    "Five-backend check 4, continuous derivative: SALD.forwardKlDerivativeCandidateContract, SALD.forwardKlDerivativeSideConditionContract, and the cycle-55 mass handoff keep the continuous Fokker-Planck/KL derivative backend visible for reused transport and schedule side conditions without promoting it.",
    "Five-backend check 5, EM interpolation: SALD.discreteForwardKlEmInterpolationSideConditionContract exposes endpoint laws, conditional drift, conditional-law density, conditional Fokker-Planck, Laplacian split, and stitched interval interfaces as source-cited obligations."
  ]
  nonGoals := [
    "Do not restate thm:forward-KL-discrete, change SALD.discreteForwardKlStatementContract, 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, residual exponent, or accumulated-error bridge in this upper packet.",
    "Do not add hidden density, absolute-continuity, endpoint, finite-log-mgf, boundary, coefficient-regularity, inverse-schedule, or stitched-interval assumptions to the theorem statement.",
    "Do not start broad SLT, concentration, measure-theory, or SDE backfill until this theorem-level route remains stable under review."
  ]
  lowerPacket := [
    "Middle should synchronize this cycle-56 upper route with conversion-windows/ASTIS-SALD-001.md and proof-obligations/ASTIS-SALD-001.md, preserving the two-way Lean/Markdown/source map.",
    "Lower should target exactly SALD.discreteForwardKlGronwallInstantiationContract / SALD.discreteForwardKlGronwallAccumulationObligation / sald.discrete_forward_kl.gronwall_accumulation over appendix.tex:526-592.",
    "First lower sub-slice: consume the existing sald.discrete_forward_kl.kl_derivative and sald.discrete_forward_kl.dv_velocity_bound interfaces to instantiate the source a(t) and b(t) in Eq. KL-derivative-7-discrete.",
    "Second lower sub-slice: keep endpoint law stitching and K(T)=KL(rho_K^eta||pi_T), K(0)=KL(rho_0||pi_0) as named side conditions rather than new theorem assumptions.",
    "Third lower sub-slice: route the linear slowdown t(s)=s/r, residual exponent, barGamma, and barDelta collection to SALD.discreteForwardKlAccumulatedErrorBridgeContract without changing the main-body coefficients."
  ]
  reviewerChecklist := [
    "SALD.discreteSaldContract lists SALD.cycle56DiscreteForwardKlSkeletonObligation while remaining contractOnly.",
    "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle56_theorem_interface_route after the cycle-51 derivative lower route and before the reusable EM/defect/DV/Gronwall nodes.",
    "SALD.saldDependenciesForLabel \"thm:forward-KL-discrete\" includes the cycle-56 upper packet, obligation, DAG node, and the explicit EM/Fokker-Planck interface names.",
    "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-56 obligation tying the discrete theorem route to the existing
source-cited EM/Fokker-Planck and Gronwall/accumulated-error interfaces. -/