AutoSamplingTheory.SALD.cycle65ForwardKlSkeletonUpperPacket

Data definition / provenance and workflow record

Meaning and type

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

def cycle65ForwardKlSkeletonUpperPacket : ForwardKlUpperPacket

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 65 upper: after the accepted cycle-64 reviewer/build gate, rewire the source-cited analytic interfaces specifically into the faithful continuous thm:forward-KL proof skeleton, matching main_body.tex:238-247 and appendix.tex:164-252 without changing constants, theorem status, proof order, or source labels.

sourceLabels : List String (explicit)

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

Ordered data items

1. thm:forward-KL
2. proof:thm:forward-KL
3. proof:thm:forward-KL:derivative
4. proof:thm:forward-KL:dv-energy
5. proof:thm:forward-KL:gronwall
6. eq:KL-derivative-0
7. eq:KL-derivative-1
8. eq:KL-derivative-2
9. lem:gronwall
10. lem:dv_variation
11. eq:LSI-KL-FI
12. eq:SALD
13. eq:FP-eq
14. def:alpha-complexity
15. proof:thm:forward-KL-discrete:conditional-fp

modeDiscipline : List String (explicit)

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

Ordered data items

1. Global phase judgment: cycle 64 passed reviewer/build, so no failed previous cycle must be recovered before cycle 65 work.
2. Global phase judgment: Phase 1 theorem-skeleton translation is stable enough for this narrow continuous forward-KL route audit, but not for broad cited-theory, SDE, disintegration, or reusable API backfill.
3. Global phase judgment: the single lower packet that best reduces the remaining continuous theorem risk is SALD.forwardKlDerivativeCandidateContract / SALD.forwardKlDerivativeObligation / sald.forward_kl.kl_derivative over appendix.tex:168-228.
4. Five-backend check: Gronwall keeps endpoint-safe differentiability/FTC, interval-integrability, endpoint, and exponent side-condition obligations; DV keeps common-space, absolute-continuity, finite-KL, selected-test measurability, finite-log-mgf, and positive-alpha interfaces; LSI/KL/FI keeps density, zero-set, admissible sqrt-density test, entropy identity, finite KL/FI, and Fisher chain-rule obligations; continuous Fokker-Planck/KL derivative keeps mass-conservation, boundary, target-transport, and schedule-calculus obligations; EM interpolation FP remains a downstream discrete sibling backend.
5. Route thm:forward-KL in the paper order appendix.tex:168-228 derivative/Fokker-Planck and time change, appendix.tex:230-241 DV velocity-energy, and appendix.tex:244-252 Gronwall endpoint/exponent display matching.
6. Preserve the exact source coefficients: the one-half Young split, dot{s}(t)^(-1), alpha in (0, alpha0], the two theorem exponent factors, and the residual alpha-complexity integral.

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, change SALD.continuousForwardKlStatementContract, or promote SALD.continuousSaldContract above contractOnly.
2. Do not prove or replace Gronwall, DV, LSI/KL/FI, the continuous Fokker-Planck/KL derivative, or the EM interpolation backend in this upper packet.
3. Do not merge exponent factors, alter source constants, add hidden endpoint/density/regularity assumptions, or use sald_version_2.tex.
4. Do not begin broad SLT, Gaussian concentration, entropy-duality, disintegration, measure-theory, or SDE backfill before this continuous route is reviewed cleanly.

lowerPacket : List String (explicit)

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

Ordered data items

1. Middle should synchronize this cycle-65 upper route with conversion-windows/ASTIS-SALD-001.md, proof-obligations/ASTIS-SALD-001.md, and the thm:forward-KL proof DAG.
2. Lower should target exactly SALD.forwardKlDerivativeCandidateContract / SALD.forwardKlDerivativeObligation / sald.forward_kl.kl_derivative over appendix.tex:168-228.
3. First lower sub-slice: appendix.tex:168-185 mass conservation, differentiating KL under the integral, SALD Fokker-Planck substitution, boundary/no-flux integration by parts, and the -FI identification.
4. Second lower sub-slice: appendix.tex:187-208 slowed target velocity, target integration by parts, Cauchy-Schwarz/Young with the exact 1/2 share, and the L2 velocity term.
5. Third lower sub-slice: appendix.tex:218-228 inverse-schedule chain rule, slowed-velocity square scaling, and dot{s}(t)*dot t(s(t))^2 = dot{s}(t)^(-1).
6. Leave LSI/KL/FI, DV finite-log-mgf, Gronwall endpoint/exponent side conditions, and EM interpolation as separate named interfaces unless their exact analytic backends compile locally.

reviewerChecklist : List String (explicit)

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

Ordered data items

1. SALD.continuousSaldContract lists SALD.cycle65ForwardKlSkeletonObligation while remaining contractOnly.
2. SALD.forwardKlProofDag contains ASTIS.SALD.forward_KL.cycle65_continuous_route and ASTIS.SALD.forward_KL.cycle65_lower_packet.kl_derivative after the cycle-64 and cycle-60 route data.
3. SALD.saldDependenciesForLabel "thm:forward-KL" includes SALD.cycle65ForwardKlSkeletonUpperPacket, SALD.cycle65ForwardKlSkeletonObligation, and the cycle-65 DAG nodes.
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 cycle65ForwardKlSkeletonUpperPacket : ForwardKlUpperPacket where
  objective := "Cycle 65 upper: after the accepted cycle-64 reviewer/build gate, rewire the source-cited analytic interfaces specifically into the faithful continuous thm:forward-KL proof skeleton, matching main_body.tex:238-247 and appendix.tex:164-252 without changing constants, theorem status, proof order, or source labels."
  sourceLabels := [
    "thm:forward-KL",
    "proof:thm:forward-KL",
    "proof:thm:forward-KL:derivative",
    "proof:thm:forward-KL:dv-energy",
    "proof:thm:forward-KL:gronwall",
    "eq:KL-derivative-0",
    "eq:KL-derivative-1",
    "eq:KL-derivative-2",
    "lem:gronwall",
    "lem:dv_variation",
    "eq:LSI-KL-FI",
    "eq:SALD",
    "eq:FP-eq",
    "def:alpha-complexity",
    "proof:thm:forward-KL-discrete:conditional-fp"
  ]
  modeDiscipline := [
    "Global phase judgment: cycle 64 passed reviewer/build, so no failed previous cycle must be recovered before cycle 65 work.",
    "Global phase judgment: Phase 1 theorem-skeleton translation is stable enough for this narrow continuous forward-KL route audit, but not for broad cited-theory, SDE, disintegration, or reusable API backfill.",
    "Global phase judgment: the single lower packet that best reduces the remaining continuous theorem risk is SALD.forwardKlDerivativeCandidateContract / SALD.forwardKlDerivativeObligation / sald.forward_kl.kl_derivative over appendix.tex:168-228.",
    "Five-backend check: Gronwall keeps endpoint-safe differentiability/FTC, interval-integrability, endpoint, and exponent side-condition obligations; DV keeps common-space, absolute-continuity, finite-KL, selected-test measurability, finite-log-mgf, and positive-alpha interfaces; LSI/KL/FI keeps density, zero-set, admissible sqrt-density test, entropy identity, finite KL/FI, and Fisher chain-rule obligations; continuous Fokker-Planck/KL derivative keeps mass-conservation, boundary, target-transport, and schedule-calculus obligations; EM interpolation FP remains a downstream discrete sibling backend.",
    "Route thm:forward-KL in the paper order appendix.tex:168-228 derivative/Fokker-Planck and time change, appendix.tex:230-241 DV velocity-energy, and appendix.tex:244-252 Gronwall endpoint/exponent display matching.",
    "Preserve the exact source coefficients: the one-half Young split, dot{s}(t)^(-1), alpha in (0, alpha0], the two theorem exponent factors, and the residual alpha-complexity integral."
  ]
  nonGoals := [
    "Do not restate thm:forward-KL, change SALD.continuousForwardKlStatementContract, or promote SALD.continuousSaldContract above contractOnly.",
    "Do not prove or replace Gronwall, DV, LSI/KL/FI, the continuous Fokker-Planck/KL derivative, or the EM interpolation backend in this upper packet.",
    "Do not merge exponent factors, alter source constants, add hidden endpoint/density/regularity assumptions, or use sald_version_2.tex.",
    "Do not begin broad SLT, Gaussian concentration, entropy-duality, disintegration, measure-theory, or SDE backfill before this continuous route is reviewed cleanly."
  ]
  lowerPacket := [
    "Middle should synchronize this cycle-65 upper route with conversion-windows/ASTIS-SALD-001.md, proof-obligations/ASTIS-SALD-001.md, and the thm:forward-KL proof DAG.",
    "Lower should target exactly SALD.forwardKlDerivativeCandidateContract / SALD.forwardKlDerivativeObligation / sald.forward_kl.kl_derivative over appendix.tex:168-228.",
    "First lower sub-slice: appendix.tex:168-185 mass conservation, differentiating KL under the integral, SALD Fokker-Planck substitution, boundary/no-flux integration by parts, and the -FI identification.",
    "Second lower sub-slice: appendix.tex:187-208 slowed target velocity, target integration by parts, Cauchy-Schwarz/Young with the exact 1/2 share, and the L2 velocity term.",
    "Third lower sub-slice: appendix.tex:218-228 inverse-schedule chain rule, slowed-velocity square scaling, and dot{s}(t)*dot t(s(t))^2 = dot{s}(t)^(-1).",
    "Leave LSI/KL/FI, DV finite-log-mgf, Gronwall endpoint/exponent side conditions, and EM interpolation as separate named interfaces unless their exact analytic backends compile locally."
  ]
  reviewerChecklist := [
    "SALD.continuousSaldContract lists SALD.cycle65ForwardKlSkeletonObligation while remaining contractOnly.",
    "SALD.forwardKlProofDag contains ASTIS.SALD.forward_KL.cycle65_continuous_route and ASTIS.SALD.forward_KL.cycle65_lower_packet.kl_derivative after the cycle-64 and cycle-60 route data.",
    "SALD.saldDependenciesForLabel \"thm:forward-KL\" includes SALD.cycle65ForwardKlSkeletonUpperPacket, SALD.cycle65ForwardKlSkeletonObligation, and the cycle-65 DAG nodes.",
    "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-65 upper obligation tying the accepted cycle-64 analytic ledger back
to the focused continuous forward-KL theorem route. -/