AutoSamplingTheory.SALD.cycle55ForwardKlSkeletonUpperPacket

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 cycle55ForwardKlSkeletonUpperPacket : 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 55 upper: after the cycle-54 analytic-interface re-check, wire those five source-cited or obligation-level interfaces back into the faithful continuous thm:forward-KL skeleton, matching main_body.tex:238-247 and appendix.tex:164-252 without changing constants, source labels, theorem status, or proof route.

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:SALD
7. eq:FP-eq
8. eq:LSI-KL-FI
9. lem:dv_variation
10. lem:gronwall
11. def:alpha-complexity
12. 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. faithfulPaper Phase 1: keep the original theorem statement from main_body.tex:238-247 and the appendix proof route from appendix.tex:164-252 fixed; sald_version_2.tex remains excluded.
2. Before assigning lower work, explicitly check the five slow backends: endpoint-safe Gronwall, common-space DV with finite KL/log-mgf, LSI/KL/FI density-test and Fisher-chain bridge, continuous Fokker-Planck/KL derivative identity, and downstream EM interpolation endpoint/conditional-law Fokker-Planck.
3. Route the proof only as derivative -> LSI -> inverse-schedule time change -> DV velocity energy -> Gronwall, preserving K(t), a(t), b(t), both 1/2 coefficients, alpha in (0,alpha0], and the two source exponent displays.
4. Endpoint, density, absolute-continuity, finite-KL/FI, finite-log-mgf, boundary, schedule, coefficient-regularity, and interval-integrability facts remain named obligations rather than hidden theorem assumptions.

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 the source exponent factors, change the coefficient (1/2)*dot{s}(t)^(-1)*alpha^(-1), or substitute an alternate entropy/transport proof route.
4. Do not begin broad SLT, concentration, measure-theory, or SDE backfill before the theorem-level route remains synchronized 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-55 upper route with conversion-windows/ASTIS-SALD-001.md, proof-obligations/ASTIS-SALD-001.md, and the thm:forward-KL proof DAG.
2. Preferred lower target remains SALD.forwardKlDerivativeCandidateContract / SALD.forwardKlDerivativeObligation / sald.forward_kl.kl_derivative over appendix.tex:168-228.
3. First lower sub-slice: mass conservation, KL differentiation under the integral, SALD Fokker-Planck substitution, boundary/no-flux integration by parts, and the -FI identification from appendix.tex:168-185.
4. Second lower sub-slice: target transport velocity, target integration by parts, Cauchy-Schwarz/Young with the exact 1/2 share, and the L2 velocity term from appendix.tex:187-208.
5. Third lower sub-slice: inverse-schedule chain rule, slowed-velocity square scaling, and dot{s}(t)*dot t(s(t))^2 = dot{s}(t)^(-1) from appendix.tex:218-228.
6. Keep LSI/KL/FI, DV finite-log-mgf, Gronwall endpoint 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.cycle55ForwardKlSkeletonObligation while remaining contractOnly.
2. SALD.forwardKlProofDag contains ASTIS.SALD.forward_KL.cycle55_continuous_skeleton_route after the cycle-54 analytic re-check and cycle-50 route nodes.
3. SALD.saldDependenciesForLabel "thm:forward-KL" includes the cycle-55 packet, obligation, DAG node, and lower-packet node.
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 cycle55ForwardKlSkeletonUpperPacket : ForwardKlUpperPacket where
  objective := "Cycle 55 upper: after the cycle-54 analytic-interface re-check, wire those five source-cited or obligation-level interfaces back into the faithful continuous thm:forward-KL skeleton, matching main_body.tex:238-247 and appendix.tex:164-252 without changing constants, source labels, theorem status, or proof route."
  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:SALD",
    "eq:FP-eq",
    "eq:LSI-KL-FI",
    "lem:dv_variation",
    "lem:gronwall",
    "def:alpha-complexity",
    "proof:thm:forward-KL-discrete:conditional-fp"
  ]
  modeDiscipline := [
    "faithfulPaper Phase 1: keep the original theorem statement from main_body.tex:238-247 and the appendix proof route from appendix.tex:164-252 fixed; sald_version_2.tex remains excluded.",
    "Before assigning lower work, explicitly check the five slow backends: endpoint-safe Gronwall, common-space DV with finite KL/log-mgf, LSI/KL/FI density-test and Fisher-chain bridge, continuous Fokker-Planck/KL derivative identity, and downstream EM interpolation endpoint/conditional-law Fokker-Planck.",
    "Route the proof only as derivative -> LSI -> inverse-schedule time change -> DV velocity energy -> Gronwall, preserving K(t), a(t), b(t), both 1/2 coefficients, alpha in (0,alpha0], and the two source exponent displays.",
    "Endpoint, density, absolute-continuity, finite-KL/FI, finite-log-mgf, boundary, schedule, coefficient-regularity, and interval-integrability facts remain named obligations rather than hidden theorem assumptions."
  ]
  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 the source exponent factors, change the coefficient (1/2)*dot{s}(t)^(-1)*alpha^(-1), or substitute an alternate entropy/transport proof route.",
    "Do not begin broad SLT, concentration, measure-theory, or SDE backfill before the theorem-level route remains synchronized under review."
  ]
  lowerPacket := [
    "Middle should synchronize this cycle-55 upper route with conversion-windows/ASTIS-SALD-001.md, proof-obligations/ASTIS-SALD-001.md, and the thm:forward-KL proof DAG.",
    "Preferred lower target remains SALD.forwardKlDerivativeCandidateContract / SALD.forwardKlDerivativeObligation / sald.forward_kl.kl_derivative over appendix.tex:168-228.",
    "First lower sub-slice: mass conservation, KL differentiation under the integral, SALD Fokker-Planck substitution, boundary/no-flux integration by parts, and the -FI identification from appendix.tex:168-185.",
    "Second lower sub-slice: target transport velocity, target integration by parts, Cauchy-Schwarz/Young with the exact 1/2 share, and the L2 velocity term from appendix.tex:187-208.",
    "Third lower sub-slice: inverse-schedule chain rule, slowed-velocity square scaling, and dot{s}(t)*dot t(s(t))^2 = dot{s}(t)^(-1) from appendix.tex:218-228.",
    "Keep LSI/KL/FI, DV finite-log-mgf, Gronwall endpoint side conditions, and EM interpolation as separate named interfaces unless their exact analytic backends compile locally."
  ]
  reviewerChecklist := [
    "SALD.continuousSaldContract lists SALD.cycle55ForwardKlSkeletonObligation while remaining contractOnly.",
    "SALD.forwardKlProofDag contains ASTIS.SALD.forward_KL.cycle55_continuous_skeleton_route after the cycle-54 analytic re-check and cycle-50 route nodes.",
    "SALD.saldDependenciesForLabel \"thm:forward-KL\" includes the cycle-55 packet, obligation, DAG node, and lower-packet node.",
    "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-55 obligation tying the re-checked analytic interfaces to the
continuous forward-KL theorem route. -/