AutoSamplingTheory.SALD.cycle32DvVariationUpperPacket

Data definition / provenance and workflow record

Meaning and type

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

def cycle32DvVariationUpperPacket : FirstAppendixVocabularyPacket

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.

Proof-closure priority check: (1) lem:gronwall remains an obligation after cycle 31 partial sublemmas; this cycle follows the requested item (2) lem:dv_variation by sharpening appendix.tex:73-79 into a precise source-cited DV interface before returning to (3) eq:LSI-KL-FI, (4) forward-KL Fokker-Planck/KL derivative, and (5) EM interpolation Fokker-Planck.

sourceLabels : List String (explicit)

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

Ordered data items

1. lem:gronwall
2. lem:dv_variation
3. eq:LSI-KL-FI
4. thm:forward-KL
5. thm:forward-KL-discrete

modeDiscipline : List String (explicit)

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

Ordered data items

1. faithfulPaper Phase 1: reproduce only appendix.tex:73-79 and the paper-cited Boucheron Corollary 4.15 interface; do not change any SALD theorem statement.
2. Keep the exact DV display KL(nu||mu)=sup_Z {E_nu[Z]-log E_mu[exp Z]} and the finite-log-mgf predicate log E_mu[exp Z] < +infty.
3. Treat DV as sourceCited until a local Mathlib/SLT port builds; downstream theorem blocks may use it only as an explicit dependency with their own common-space, measurability, absolute-continuity, and finite-log-mgf witnesses.
4. Do not spend this cycle on source-index rebaseline unless a reviewer identifies a blocking source-anchor defect.

nonGoals : List String (explicit)

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

Ordered data items

1. Do not mark probability.dv_variational_formula, SALD.dvContract, or lem:dv_variation formalized.
2. Do not replace DV with Pinsker, Talagrand, LSI, Girsanov, or any other entropy inequality.
3. Do not reopen broad transcript expansion, polished article export, PI velocity-norm work, LSI density-test proof search, forward-KL derivative proof search, or EM interpolation proof search in this upper packet.
4. Do not add hidden boundedness, smoothness, or exponential-integrability assumptions to the source theorem statements.

lowerPacket : List String (explicit)

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

Ordered data items

1. Middle must keep two-way Lean/Markdown/LaTeX synchronization for appendix.tex:73-79, AutoSamplingTheory/Probability.lean, AutoSamplingTheory/SALD.lean, conversion-windows/ASTIS-SALD-001.md, and proof-obligations/ASTIS-SALD-001.md.
2. Lower should first check whether the current local Mathlib state already exposes a usable entropy-duality theorem. If not, do not axiomatize it; refine the source-cited interface around dvVariationalFormulaInterface saldDvVariationSource and SALD.saldDvFiniteLogMgfContract.
3. Target exactly the DV interface slice: same-space probability measures mu and nu, real measurable tests Z, finite predicate log E_mu[exp Z] < +infty, equality as a supremum over such tests, and the one-sided consequence E_nu[Z] <= KL(nu||mu)+log E_mu[exp Z].
4. Keep theorem-specific SALD applications as obligations: forward-KL, discrete forward-KL, and general moving-target blocks must still supply common-space, absolute-continuity, measurability, finite-log-mgf, alpha0-to-alpha monotonicity, and positive-alpha scaling.

reviewerChecklist : List String (explicit)

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

Ordered data items

1. SALD.dvContract and SALD.saldStatusForLabel "lem:dv_variation" still report ProofStatus.sourceCited.
2. AutoSamplingTheory/Probability.lean contains only contract/interface data for DV, not an axiom or fake proof closure.
3. SALD.saldDependenciesForLabel "lem:dv_variation" lists dvVariationalFormulaInterface and the cycle-32 DV interface obligation as explicit dependencies.
4. The conversion window, proof-obligation ledger, cited-results audit, and dialogue board all classify cycle 32 as source-cited DV interface refinement, not proof formalization.
5. The mandatory gate python3 tools/astis.py check passes and the forbidden-proof scan remains clean.

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 cycle32DvVariationUpperPacket : FirstAppendixVocabularyPacket where
  objective := "Proof-closure priority check: (1) lem:gronwall remains an obligation after cycle 31 partial sublemmas; this cycle follows the requested item (2) lem:dv_variation by sharpening appendix.tex:73-79 into a precise source-cited DV interface before returning to (3) eq:LSI-KL-FI, (4) forward-KL Fokker-Planck/KL derivative, and (5) EM interpolation Fokker-Planck."
  sourceLabels := [
    "lem:gronwall",
    "lem:dv_variation",
    "eq:LSI-KL-FI",
    "thm:forward-KL",
    "thm:forward-KL-discrete"
  ]
  modeDiscipline := [
    "faithfulPaper Phase 1: reproduce only appendix.tex:73-79 and the paper-cited Boucheron Corollary 4.15 interface; do not change any SALD theorem statement.",
    "Keep the exact DV display KL(nu||mu)=sup_Z {E_nu[Z]-log E_mu[exp Z]} and the finite-log-mgf predicate log E_mu[exp Z] < +infty.",
    "Treat DV as sourceCited until a local Mathlib/SLT port builds; downstream theorem blocks may use it only as an explicit dependency with their own common-space, measurability, absolute-continuity, and finite-log-mgf witnesses.",
    "Do not spend this cycle on source-index rebaseline unless a reviewer identifies a blocking source-anchor defect."
  ]
  nonGoals := [
    "Do not mark probability.dv_variational_formula, SALD.dvContract, or lem:dv_variation formalized.",
    "Do not replace DV with Pinsker, Talagrand, LSI, Girsanov, or any other entropy inequality.",
    "Do not reopen broad transcript expansion, polished article export, PI velocity-norm work, LSI density-test proof search, forward-KL derivative proof search, or EM interpolation proof search in this upper packet.",
    "Do not add hidden boundedness, smoothness, or exponential-integrability assumptions to the source theorem statements."
  ]
  lowerPacket := [
    "Middle must keep two-way Lean/Markdown/LaTeX synchronization for appendix.tex:73-79, AutoSamplingTheory/Probability.lean, AutoSamplingTheory/SALD.lean, conversion-windows/ASTIS-SALD-001.md, and proof-obligations/ASTIS-SALD-001.md.",
    "Lower should first check whether the current local Mathlib state already exposes a usable entropy-duality theorem. If not, do not axiomatize it; refine the source-cited interface around dvVariationalFormulaInterface saldDvVariationSource and SALD.saldDvFiniteLogMgfContract.",
    "Target exactly the DV interface slice: same-space probability measures mu and nu, real measurable tests Z, finite predicate log E_mu[exp Z] < +infty, equality as a supremum over such tests, and the one-sided consequence E_nu[Z] <= KL(nu||mu)+log E_mu[exp Z].",
    "Keep theorem-specific SALD applications as obligations: forward-KL, discrete forward-KL, and general moving-target blocks must still supply common-space, absolute-continuity, measurability, finite-log-mgf, alpha0-to-alpha monotonicity, and positive-alpha scaling."
  ]
  reviewerChecklist := [
    "SALD.dvContract and SALD.saldStatusForLabel \"lem:dv_variation\" still report ProofStatus.sourceCited.",
    "AutoSamplingTheory/Probability.lean contains only contract/interface data for DV, not an axiom or fake proof closure.",
    "SALD.saldDependenciesForLabel \"lem:dv_variation\" lists dvVariationalFormulaInterface and the cycle-32 DV interface obligation as explicit dependencies.",
    "The conversion window, proof-obligation ledger, cited-results audit, and dialogue board all classify cycle 32 as source-cited DV interface refinement, not proof formalization.",
    "The mandatory gate python3 tools/astis.py check passes and the forbidden-proof scan remains clean."
  ]
  status := ProofStatus.obligation

/-- Cycle-32 source-cited interface obligation for the cited DV formula. -/