AutoSamplingTheory.SALD.cycle29FirstAppendixMiddleAuditContract

Data definition / provenance and workflow record

Meaning and type

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

def cycle29FirstAppendixMiddleAuditContract :
    FirstAppendixMiddleAuditContract

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.

sourceIndexPath : String (explicit)

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

research-wiki/source-index/SALD_original.jsonl

focusLabels : 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. def:PI
4. eq:LSI-KL-FI

sourceReadWindows : List String (explicit)

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

Ordered data items

1. appendix.tex:47-71: Gronwall statement and integrating-factor proof; unchanged status guard for this cycle.
2. appendix.tex:73-79: Donsker--Varadhan formula cited as Boucheron Cor. 4.15; unchanged source-cited dependency.
3. appendix.tex:86-151: PI definition plus the weighted Sobolev/Riesz velocity-norm route; unchanged contract-only definition and backend obligation.
4. main_body.tex:202-215: LSI definition, rho << pi, phi=sqrt(rho/pi), KL/FI vocabulary, and the coefficient 1/(2*C_LSI), with the selected lower sub-slice at lines 208-215.

sourceStepMap : List String (explicit)

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

Ordered data items

1. lem:gronwall: keep dK/dt <= -a_t*K_t+b_t and the endpoint exponential display as existing Gronwall obligations; no Gronwall proof search is selected in cycle 29.
2. lem:dv_variation: keep the cited variational equality over finite-log-mgf random variables; local theorem instantiations still owe common-space, measurability, and finite-log-mgf witnesses.
3. def:PI: keep lines 86-94 as the PI definition and lines 96-151 as the separate velocity-norm Sobolev/Riesz backend; no PI-to-LSI route is introduced.
4. eq:LSI-KL-FI: lines 208-215 require rho << pi, the Radon-Nikodym density r=rho/pi, the LSI test phi=sqrt(r), normalization int phi^2 dpi=1, entropy identity with KL, FI chain rule with factor 1/4, and the final coefficient 1/(2*C_LSI).

leanStepMap : List String (explicit)

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

Ordered data items

1. lem:gronwall -> SALD.saldGronwallCandidateContract, SALD.saldGronwallEndpointCalculusContract, SALD.saldGronwallExponentRewriteContract, and existing sald.gronwall.* obligations.
2. lem:dv_variation -> SALD.dvContract, SALD.saldDvFiniteLogMgfContract, probability.dv_variational_formula, and sald.dv_variation.finite_log_mgf_interface.
3. def:PI -> SALD.saldPIContract, SALD.piDefinitionContract, SALD.saldPiVelocityNormDependencyContract, and sald.pi.velocity_norm_backend.
4. eq:LSI-KL-FI -> SALD.saldKLContract, SALD.saldFIContract, SALD.saldLSIContract, SALD.saldLsiKlFiBridgeContract, SALD.saldLsiKlFiDensityTestContract, SALD.lsiKlFiDensityTestObligation, SALD.cycle29LsiKlFiDensityTestMiddleObligation, sald.lsi_kl_fi.density_test_interface, and probability.lsi_to_kl_fi.

citedResultMap : List String (explicit)

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

Ordered data items

1. DV remains the only external cited result in the first-layer source window; no SLT theorem is imported or promoted.
2. The LSI/KL/FI bridge is local measure and density-test analysis, not an SLT reuse result and not a theorem-level VA-SALD proof.
3. Gronwall and PI remain local real-analysis/Sobolev obligations outside this lower slice.

obligationMap : List String (explicit)

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

Ordered data items

1. sald.lsi_kl_fi.density_test_interface first sub-slice: expose the Radon-Nikodym density ratio r=d rho/d pi and normalization int r dpi=1 for probability densities.
2. sald.lsi_kl_fi.density_test_interface admissibility sub-slice: justify phi=sqrt(r) as a smooth/admissible LSI test function or record an approximation/closure lemma.
3. sald.lsi_kl_fi.density_test_interface identity sub-slice: rewrite integral phi^2 log(phi^2) dpi as KL(rho||pi) and integral ||nabla phi||^2 dpi as (1/4)*FI(rho||pi), preserving finite KL/FI hypotheses and zero-density conventions.
4. sald.lsi_kl_fi.density_test_interface coefficient sub-slice: combine the source LSI factor 2/C_LSI with the FI chain-rule factor 1/4 to obtain exactly FI/(2*C_LSI); SALD.lsiKlFiCoefficientAuditScalar formalizes only this scalar algebra, while the analytic inputs remain obligations.

lowerPacket : List String (explicit)

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

Ordered data items

1. Target exactly SALD.saldLsiKlFiDensityTestContract / SALD.lsiKlFiDensityTestObligation / sald.lsi_kl_fi.density_test_interface.
2. Start with main_body.tex:208-215: rho << pi, r=rho/pi, phi=sqrt(r), normalization, entropy rewrite, FI chain rule, finite KL/FI interfaces, and coefficient audit.
3. If a backend is missing, refine SALD.cycle29LsiKlFiDensityTestMiddleObligation with the exact density, admissibility, approximation, zero-density, or chain-rule gap.
4. Do not reopen Gronwall, DV, PI velocity-norm, forward-KL, guided, general VA-SALD, unified, or discrete theorem proof search in this lower attempt.

reviewerChecklist : List String (explicit)

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

Ordered data items

1. SALD_original.jsonl indexes lem:gronwall, lem:dv_variation, def:PI, lem:velocity-norm-bound, and eq:LSI-KL-FI from the original files while excluding sald_version_2.tex.
2. SALD.saldFirstProofDag dependencies for the four first-layer labels include SALD.cycle29FirstAppendixMiddleAuditContract.
3. eq:LSI-KL-FI dependencies include SALD.cycle29LsiKlFiDensityTestMiddleObligation and still report LSI/KL/FI as an obligation.
4. Conversion window, proof-obligation ledger, SLT audit, and dialogue board classify cycle 29 middle as source-to-Lean synchronization, not analytic proof closure.

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 cycle29FirstAppendixMiddleAuditContract :
    FirstAppendixMiddleAuditContract where
  sourceIndexPath := "research-wiki/source-index/SALD_original.jsonl"
  focusLabels := [
    "lem:gronwall",
    "lem:dv_variation",
    "def:PI",
    "eq:LSI-KL-FI"
  ]
  sourceReadWindows := [
    "appendix.tex:47-71: Gronwall statement and integrating-factor proof; unchanged status guard for this cycle.",
    "appendix.tex:73-79: Donsker--Varadhan formula cited as Boucheron Cor. 4.15; unchanged source-cited dependency.",
    "appendix.tex:86-151: PI definition plus the weighted Sobolev/Riesz velocity-norm route; unchanged contract-only definition and backend obligation.",
    "main_body.tex:202-215: LSI definition, rho << pi, phi=sqrt(rho/pi), KL/FI vocabulary, and the coefficient 1/(2*C_LSI), with the selected lower sub-slice at lines 208-215."
  ]
  sourceStepMap := [
    "lem:gronwall: keep dK/dt <= -a_t*K_t+b_t and the endpoint exponential display as existing Gronwall obligations; no Gronwall proof search is selected in cycle 29.",
    "lem:dv_variation: keep the cited variational equality over finite-log-mgf random variables; local theorem instantiations still owe common-space, measurability, and finite-log-mgf witnesses.",
    "def:PI: keep lines 86-94 as the PI definition and lines 96-151 as the separate velocity-norm Sobolev/Riesz backend; no PI-to-LSI route is introduced.",
    "eq:LSI-KL-FI: lines 208-215 require rho << pi, the Radon-Nikodym density r=rho/pi, the LSI test phi=sqrt(r), normalization int phi^2 dpi=1, entropy identity with KL, FI chain rule with factor 1/4, and the final coefficient 1/(2*C_LSI)."
  ]
  leanStepMap := [
    "lem:gronwall -> SALD.saldGronwallCandidateContract, SALD.saldGronwallEndpointCalculusContract, SALD.saldGronwallExponentRewriteContract, and existing sald.gronwall.* obligations.",
    "lem:dv_variation -> SALD.dvContract, SALD.saldDvFiniteLogMgfContract, probability.dv_variational_formula, and sald.dv_variation.finite_log_mgf_interface.",
    "def:PI -> SALD.saldPIContract, SALD.piDefinitionContract, SALD.saldPiVelocityNormDependencyContract, and sald.pi.velocity_norm_backend.",
    "eq:LSI-KL-FI -> SALD.saldKLContract, SALD.saldFIContract, SALD.saldLSIContract, SALD.saldLsiKlFiBridgeContract, SALD.saldLsiKlFiDensityTestContract, SALD.lsiKlFiDensityTestObligation, SALD.cycle29LsiKlFiDensityTestMiddleObligation, sald.lsi_kl_fi.density_test_interface, and probability.lsi_to_kl_fi."
  ]
  citedResultMap := [
    "DV remains the only external cited result in the first-layer source window; no SLT theorem is imported or promoted.",
    "The LSI/KL/FI bridge is local measure and density-test analysis, not an SLT reuse result and not a theorem-level VA-SALD proof.",
    "Gronwall and PI remain local real-analysis/Sobolev obligations outside this lower slice."
  ]
  obligationMap := [
    "sald.lsi_kl_fi.density_test_interface first sub-slice: expose the Radon-Nikodym density ratio r=d rho/d pi and normalization int r dpi=1 for probability densities.",
    "sald.lsi_kl_fi.density_test_interface admissibility sub-slice: justify phi=sqrt(r) as a smooth/admissible LSI test function or record an approximation/closure lemma.",
    "sald.lsi_kl_fi.density_test_interface identity sub-slice: rewrite integral phi^2 log(phi^2) dpi as KL(rho||pi) and integral ||nabla phi||^2 dpi as (1/4)*FI(rho||pi), preserving finite KL/FI hypotheses and zero-density conventions.",
    "sald.lsi_kl_fi.density_test_interface coefficient sub-slice: combine the source LSI factor 2/C_LSI with the FI chain-rule factor 1/4 to obtain exactly FI/(2*C_LSI); SALD.lsiKlFiCoefficientAuditScalar formalizes only this scalar algebra, while the analytic inputs remain obligations."
  ]
  lowerPacket := [
    "Target exactly SALD.saldLsiKlFiDensityTestContract / SALD.lsiKlFiDensityTestObligation / sald.lsi_kl_fi.density_test_interface.",
    "Start with main_body.tex:208-215: rho << pi, r=rho/pi, phi=sqrt(r), normalization, entropy rewrite, FI chain rule, finite KL/FI interfaces, and coefficient audit.",
    "If a backend is missing, refine SALD.cycle29LsiKlFiDensityTestMiddleObligation with the exact density, admissibility, approximation, zero-density, or chain-rule gap.",
    "Do not reopen Gronwall, DV, PI velocity-norm, forward-KL, guided, general VA-SALD, unified, or discrete theorem proof search in this lower attempt."
  ]
  reviewerChecklist := [
    "SALD_original.jsonl indexes lem:gronwall, lem:dv_variation, def:PI, lem:velocity-norm-bound, and eq:LSI-KL-FI from the original files while excluding sald_version_2.tex.",
    "SALD.saldFirstProofDag dependencies for the four first-layer labels include SALD.cycle29FirstAppendixMiddleAuditContract.",
    "eq:LSI-KL-FI dependencies include SALD.cycle29LsiKlFiDensityTestMiddleObligation and still report LSI/KL/FI as an obligation.",
    "Conversion window, proof-obligation ledger, SLT audit, and dialogue board classify cycle 29 middle as source-to-Lean synchronization, not analytic proof closure."
  ]
  status := ProofStatus.obligation

/-- Cycle-29 middle obligation for the selected LSI/KL/FI density-test sub-slice. -/