AutoSamplingTheory.SALD.cycle43LsiKlFiUpperPacket

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 cycle43LsiKlFiUpperPacket : 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 before assigning lower work: (1) lem:gronwall has endpoint-safe local calculus progress from cycle 41 but remains an obligation until the paper's concise differentiability hypothesis is connected to the closed-interval or absolute-continuity backend; (2) lem:dv_variation has cycle 42 selected scaled-test finite-mgf and one-sided energy sublemmas while the Boucheron supremum equality remains sourceCited; (3) this cycle therefore selects eq:LSI-KL-FI, specifically the density/test-function and coefficient sublemmas in main_body.tex:202-215; (4) the forward-KL Fokker-Planck/KL derivative identity and (5) the EM interpolation Fokker-Planck backend remain later closure targets.

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: use only the original main_body.tex:202-215 LSI, KL, and FI bridge; sald_version_2.tex remains excluded.
2. Keep the source theorem fixed: pi satisfies LSI with constant C_LSI > 0, rho << pi, phi=sqrt(rho/pi), KL and FI are the displayed integrals, and the comparison is exactly KL(rho||pi) <= FI(rho||pi)/(2*C_LSI).
3. Build proof-producing code only for source-shaped density/test-function, normalization, entropy-identity, Fisher-chain, or coefficient sublemmas; leave any large analytic theorem as a precise source-cited or obligation interface.
4. Keep probability.lsi_to_kl_fi, SALD.lsiKlFiDensityTestObligation, and SALD.saldStatusForLabel "eq:LSI-KL-FI" below formalized status until the full Radon-Nikodym, admissibility, integral, and Sobolev-chain backend compiles.

nonGoals : List String (explicit)

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

Ordered data items

1. Do not run a source-index rebaseline unless a reviewer identifies a blocking source-anchor defect.
2. Do not reopen Gronwall, Donsker-Varadhan, forward-KL derivative, EM interpolation, PI velocity-norm, guided, general VA-SALD, unified, or accumulated-error proof search in this upper packet.
3. Do not replace the source LSI route with PI, Pinsker, Talagrand, transport, or theorem-level forward-KL reasoning.
4. Do not add hidden smoothness, positivity, finite-KL, finite-FI, boundedness, approximation, or zero-density hypotheses to thm:forward-KL or any downstream theorem statement.
5. Do not consume the cycle-29 coefficient audit or cycle-33/38 scalar bridges unless the analytic premises are explicit assumptions, compiled lemmas, or source-cited interfaces.

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 main_body.tex:202-215, AutoSamplingTheory/Probability.lean, AutoSamplingTheory/SALD.lean, conversion-windows/ASTIS-SALD-001.md, and proof-obligations/ASTIS-SALD-001.md.
2. Target exactly SALD.saldLsiKlFiDensityTestContract, SALD.lsiKlFiDensityTestObligation, and sald.lsi_kl_fi.density_test_interface; do not change theorem statements.
3. First proof-producing lower attempt: add a narrow lemma or interface for one of the remaining measure-level inputs after cycle 38: Radon-Nikodym density normalization int r d pi=1, entropy integral transport integral r log r d pi = KL(rho||pi), admissibility or approximation of phi=sqrt(r), zero-density convention, or the vector/integral Fisher chain rule int ||nabla sqrt(r)||^2 d pi = (1/4)*FI(rho||pi).
4. If the full Sobolev or measure-theoretic backend is too large for local Mathlib, record a precise source-cited theorem interface naming absolute continuity, nonnegative density ratio, admissible sqrt test or approximation, finite KL/FI, zero-density handling, and the exact integral identities; keep the interface status below formalized.
5. After any lower proof, translate the accepted Lean declaration back into the conversion window and proof-obligation ledger, and keep SLT audit entries at no-slt-import unless an actual built dependency is introduced.

reviewerChecklist : List String (explicit)

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

Ordered data items

1. The packet explicitly checks the closure order (Gronwall, DV, LSI/KL/FI, forward-KL derivative, EM interpolation) and selects item (3), eq:LSI-KL-FI.
2. Any new Lean proof is tied to main_body.tex:202-215 and is only a local density/test-function, normalization, entropy, Fisher-chain, or coefficient sublemma; no downstream theorem receives hidden assumptions.
3. SALD.lsiKlFiVocabularyContract, SALD.lsiKlFiDensityTestObligation, probability.lsi_to_kl_fi, and SALD.saldStatusForLabel "eq:LSI-KL-FI" remain ProofStatus.obligation.
4. No axiom, sorry, admit, Prop := True, := trivial, source-file drift, SLT import, or alternate non-LSI proof route is introduced.
5. The mandatory gate python3 tools/astis.py check passes.

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 cycle43LsiKlFiUpperPacket : FirstAppendixVocabularyPacket where
  objective := "Proof-closure priority check before assigning lower work: (1) lem:gronwall has endpoint-safe local calculus progress from cycle 41 but remains an obligation until the paper's concise differentiability hypothesis is connected to the closed-interval or absolute-continuity backend; (2) lem:dv_variation has cycle 42 selected scaled-test finite-mgf and one-sided energy sublemmas while the Boucheron supremum equality remains sourceCited; (3) this cycle therefore selects eq:LSI-KL-FI, specifically the density/test-function and coefficient sublemmas in main_body.tex:202-215; (4) the forward-KL Fokker-Planck/KL derivative identity and (5) the EM interpolation Fokker-Planck backend remain later closure targets."
  sourceLabels := [
    "lem:gronwall",
    "lem:dv_variation",
    "eq:LSI-KL-FI",
    "thm:forward-KL",
    "thm:forward-KL-discrete"
  ]
  modeDiscipline := [
    "faithfulPaper Phase 1: use only the original main_body.tex:202-215 LSI, KL, and FI bridge; sald_version_2.tex remains excluded.",
    "Keep the source theorem fixed: pi satisfies LSI with constant C_LSI > 0, rho << pi, phi=sqrt(rho/pi), KL and FI are the displayed integrals, and the comparison is exactly KL(rho||pi) <= FI(rho||pi)/(2*C_LSI).",
    "Build proof-producing code only for source-shaped density/test-function, normalization, entropy-identity, Fisher-chain, or coefficient sublemmas; leave any large analytic theorem as a precise source-cited or obligation interface.",
    "Keep probability.lsi_to_kl_fi, SALD.lsiKlFiDensityTestObligation, and SALD.saldStatusForLabel \"eq:LSI-KL-FI\" below formalized status until the full Radon-Nikodym, admissibility, integral, and Sobolev-chain backend compiles."
  ]
  nonGoals := [
    "Do not run a source-index rebaseline unless a reviewer identifies a blocking source-anchor defect.",
    "Do not reopen Gronwall, Donsker-Varadhan, forward-KL derivative, EM interpolation, PI velocity-norm, guided, general VA-SALD, unified, or accumulated-error proof search in this upper packet.",
    "Do not replace the source LSI route with PI, Pinsker, Talagrand, transport, or theorem-level forward-KL reasoning.",
    "Do not add hidden smoothness, positivity, finite-KL, finite-FI, boundedness, approximation, or zero-density hypotheses to thm:forward-KL or any downstream theorem statement.",
    "Do not consume the cycle-29 coefficient audit or cycle-33/38 scalar bridges unless the analytic premises are explicit assumptions, compiled lemmas, or source-cited interfaces."
  ]
  lowerPacket := [
    "Middle must keep two-way Lean/Markdown/LaTeX synchronization for main_body.tex:202-215, AutoSamplingTheory/Probability.lean, AutoSamplingTheory/SALD.lean, conversion-windows/ASTIS-SALD-001.md, and proof-obligations/ASTIS-SALD-001.md.",
    "Target exactly SALD.saldLsiKlFiDensityTestContract, SALD.lsiKlFiDensityTestObligation, and sald.lsi_kl_fi.density_test_interface; do not change theorem statements.",
    "First proof-producing lower attempt: add a narrow lemma or interface for one of the remaining measure-level inputs after cycle 38: Radon-Nikodym density normalization int r d pi=1, entropy integral transport integral r log r d pi = KL(rho||pi), admissibility or approximation of phi=sqrt(r), zero-density convention, or the vector/integral Fisher chain rule int ||nabla sqrt(r)||^2 d pi = (1/4)*FI(rho||pi).",
    "If the full Sobolev or measure-theoretic backend is too large for local Mathlib, record a precise source-cited theorem interface naming absolute continuity, nonnegative density ratio, admissible sqrt test or approximation, finite KL/FI, zero-density handling, and the exact integral identities; keep the interface status below formalized.",
    "After any lower proof, translate the accepted Lean declaration back into the conversion window and proof-obligation ledger, and keep SLT audit entries at no-slt-import unless an actual built dependency is introduced."
  ]
  reviewerChecklist := [
    "The packet explicitly checks the closure order (Gronwall, DV, LSI/KL/FI, forward-KL derivative, EM interpolation) and selects item (3), eq:LSI-KL-FI.",
    "Any new Lean proof is tied to main_body.tex:202-215 and is only a local density/test-function, normalization, entropy, Fisher-chain, or coefficient sublemma; no downstream theorem receives hidden assumptions.",
    "SALD.lsiKlFiVocabularyContract, SALD.lsiKlFiDensityTestObligation, probability.lsi_to_kl_fi, and SALD.saldStatusForLabel \"eq:LSI-KL-FI\" remain ProofStatus.obligation.",
    "No axiom, sorry, admit, Prop := True, := trivial, source-file drift, SLT import, or alternate non-LSI proof route is introduced.",
    "The mandatory gate python3 tools/astis.py check passes."
  ]
  status := ProofStatus.obligation

/-- Cycle-43 upper workflow obligation for the LSI/KL/FI density-test backend. -/