AutoSamplingTheory.SALD.cycle35DiscreteForwardKlEmFpUpperPacket

Data definition / provenance and workflow record

Meaning and type

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

def cycle35DiscreteForwardKlEmFpUpperPacket : DiscreteForwardKlUpperPacket

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 open after cycle 31 local real-analysis sublemmas, (2) lem:dv_variation remains source-cited with cycle 32 scalar consequences, (3) eq:LSI-KL-FI remains open after cycle 33 density-test scalar lemmas, and (4) the continuous forward-KL derivative has cycle 34 scalar handoffs but still depends on analytic Fokker-Planck inputs; therefore this cycle follows item (5), the Euler-Maruyama interpolation Fokker-Planck backend for appendix.tex:260-385.

sourceLabels : List String (explicit)

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

Ordered data items

1. thm:forward-KL-discrete
2. eq:frozen_interp_terminal_disc_prop_additive_final
3. proof:thm:forward-KL-discrete:em-interpolation-fp
4. lem:frozen_delta_cross_lip_sald
5. eq:LSI-KL-FI
6. lem:dv_variation
7. lem:gronwall

modeDiscipline : List String (explicit)

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

Ordered data items

1. faithfulPaper: use main_body.tex:299-323 and appendix.tex:260-385 from the original source root; sald_version_2.tex remains excluded.
2. Preserve the source order: define the frozen interpolation hat X_s, use endpoint laws on [s_k,s_{k+1}], define bar b_{k,s} by conditional expectation, invoke partial_s hat rho_s = -div(hat rho_s*bar b_{k,s}) + Delta hat rho_s, split the Laplacian relative to tilde pi_s, then hand off to the KL derivative block.
3. Keep thm:forward-KL-discrete, t(s)=s/r, Gamma, Delta, barGamma, barDelta, alpha, alpha', eta, r, and the step-size condition unchanged.
4. Treat endpoint laws, conditional expectation/disintegration, density/positivity, Fokker-Planck, Laplacian split, boundary integration by parts, stitched regularity, LSI, DV, and Gronwall as obligations unless a compiled local proof replaces exactly one named interface.

nonGoals : List String (explicit)

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

Ordered data items

1. Do not rebaseline the source index or expand the whole discrete theorem transcript unless a reviewer finds a blocking source-anchor defect.
2. Do not prove or restate thm:forward-KL-discrete in this packet.
3. Do not jump to frozen Gamma/Delta, DV velocity, Gronwall accumulation, or accumulated-error collection before the EM conditional-FP backend is narrowed.
4. Do not import or mark an SLT one_step_discretization theorem formalized; it remains at most a route reference.
5. Do not close the conditional Fokker-Planck backend by adding theorem-level smoothness, density, disintegration, or boundary hypotheses.

lowerPacket : List String (explicit)

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

Ordered data items

1. Target exactly SALD.discreteForwardKlEmInterpolationSideConditionContract and SALD.discreteForwardKlEmConditionalFpObligation, registered by SALD.cycle35DiscreteForwardKlEmFpUpperObligation / sald.discrete_forward_kl.cycle35_em_fp_upper.
2. Middle must keep the two-way source map synchronized for appendix.tex:260-385: endpoint interpolation laws, bar b_{k,s} conditional drift, conditional-drift Fokker-Planck equation, Laplacian split relative to tilde pi_s, and KL-derivative handoff.
3. Lower should attempt one proof-producing Lean interface first: either an endpoint-law algebra lemma for the frozen interpolation at s=s_k and s=s_{k+1}, or a narrow conditional-drift measurability/density interface for bar b_{k,s}.
4. If the analytic Fokker-Planck theorem is too large for local Mathlib, create a precise source-cited theorem interface depending on the existing conditional-drift density obligation; keep its status below formalized.
5. Keep endpoint stitching, frozen one-step Gamma/Delta estimates, LSI, DV, Gronwall, and accumulated-error constants as sibling obligations outside the lower attempt.

reviewerChecklist : List String (explicit)

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

Ordered data items

1. SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle35_em_fp_upper before derivative, DV, and Gronwall blocks.
2. SALD.saldDependenciesForLabel "thm:forward-KL-discrete" includes SALD.cycle35DiscreteForwardKlEmFpUpperPacket, sald.discrete_forward_kl.cycle35_em_fp_upper, and the existing EM endpoint/conditional-FP obligations.
3. SALD.discreteSaldContract lists SALD.cycle35DiscreteForwardKlEmFpUpperObligation without removing cycle-15 EM endpoint, conditional-drift density, conditional-FP, or stitched-interval obligations.
4. No theorem constants, source files, source route, or analytic dependency statuses are changed.
5. The mandatory source-index and ASTIS checks pass, and no fake proof closure pattern appears.

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 cycle35DiscreteForwardKlEmFpUpperPacket : DiscreteForwardKlUpperPacket where
  objective := "Proof-closure priority check: (1) lem:gronwall remains open after cycle 31 local real-analysis sublemmas, (2) lem:dv_variation remains source-cited with cycle 32 scalar consequences, (3) eq:LSI-KL-FI remains open after cycle 33 density-test scalar lemmas, and (4) the continuous forward-KL derivative has cycle 34 scalar handoffs but still depends on analytic Fokker-Planck inputs; therefore this cycle follows item (5), the Euler-Maruyama interpolation Fokker-Planck backend for appendix.tex:260-385."
  sourceLabels := [
    "thm:forward-KL-discrete",
    "eq:frozen_interp_terminal_disc_prop_additive_final",
    "proof:thm:forward-KL-discrete:em-interpolation-fp",
    "lem:frozen_delta_cross_lip_sald",
    "eq:LSI-KL-FI",
    "lem:dv_variation",
    "lem:gronwall"
  ]
  modeDiscipline := [
    "faithfulPaper: use main_body.tex:299-323 and appendix.tex:260-385 from the original source root; sald_version_2.tex remains excluded.",
    "Preserve the source order: define the frozen interpolation hat X_s, use endpoint laws on [s_k,s_{k+1}], define bar b_{k,s} by conditional expectation, invoke partial_s hat rho_s = -div(hat rho_s*bar b_{k,s}) + Delta hat rho_s, split the Laplacian relative to tilde pi_s, then hand off to the KL derivative block.",
    "Keep thm:forward-KL-discrete, t(s)=s/r, Gamma, Delta, barGamma, barDelta, alpha, alpha', eta, r, and the step-size condition unchanged.",
    "Treat endpoint laws, conditional expectation/disintegration, density/positivity, Fokker-Planck, Laplacian split, boundary integration by parts, stitched regularity, LSI, DV, and Gronwall as obligations unless a compiled local proof replaces exactly one named interface."
  ]
  nonGoals := [
    "Do not rebaseline the source index or expand the whole discrete theorem transcript unless a reviewer finds a blocking source-anchor defect.",
    "Do not prove or restate thm:forward-KL-discrete in this packet.",
    "Do not jump to frozen Gamma/Delta, DV velocity, Gronwall accumulation, or accumulated-error collection before the EM conditional-FP backend is narrowed.",
    "Do not import or mark an SLT one_step_discretization theorem formalized; it remains at most a route reference.",
    "Do not close the conditional Fokker-Planck backend by adding theorem-level smoothness, density, disintegration, or boundary hypotheses."
  ]
  lowerPacket := [
    "Target exactly SALD.discreteForwardKlEmInterpolationSideConditionContract and SALD.discreteForwardKlEmConditionalFpObligation, registered by SALD.cycle35DiscreteForwardKlEmFpUpperObligation / sald.discrete_forward_kl.cycle35_em_fp_upper.",
    "Middle must keep the two-way source map synchronized for appendix.tex:260-385: endpoint interpolation laws, bar b_{k,s} conditional drift, conditional-drift Fokker-Planck equation, Laplacian split relative to tilde pi_s, and KL-derivative handoff.",
    "Lower should attempt one proof-producing Lean interface first: either an endpoint-law algebra lemma for the frozen interpolation at s=s_k and s=s_{k+1}, or a narrow conditional-drift measurability/density interface for bar b_{k,s}.",
    "If the analytic Fokker-Planck theorem is too large for local Mathlib, create a precise source-cited theorem interface depending on the existing conditional-drift density obligation; keep its status below formalized.",
    "Keep endpoint stitching, frozen one-step Gamma/Delta estimates, LSI, DV, Gronwall, and accumulated-error constants as sibling obligations outside the lower attempt."
  ]
  reviewerChecklist := [
    "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle35_em_fp_upper before derivative, DV, and Gronwall blocks.",
    "SALD.saldDependenciesForLabel \"thm:forward-KL-discrete\" includes SALD.cycle35DiscreteForwardKlEmFpUpperPacket, sald.discrete_forward_kl.cycle35_em_fp_upper, and the existing EM endpoint/conditional-FP obligations.",
    "SALD.discreteSaldContract lists SALD.cycle35DiscreteForwardKlEmFpUpperObligation without removing cycle-15 EM endpoint, conditional-drift density, conditional-FP, or stitched-interval obligations.",
    "No theorem constants, source files, source route, or analytic dependency statuses are changed.",
    "The mandatory source-index and ASTIS checks pass, and no fake proof closure pattern appears."
  ]
  status := ProofStatus.obligation

/-- Cycle-35 middle packet for the EM interpolation Fokker--Planck sprint.

This translates `appendix.tex:260-385` into lower-ready Lean targets while
keeping the analytic endpoint-law and conditional-drift Fokker--Planck
backends open.  The compiled endpoint and divergence regrouping lemmas are
local algebra only; they do not prove stochastic laws, disintegration,
density regularity, or integration by parts.
-/