AutoSamplingTheory.SALD.cycle61DiscreteForwardKlSkeletonUpperPacket

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 cycle61DiscreteForwardKlSkeletonUpperPacket :
    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.

Cycle 61 upper: recover the interrupted cycle-56 thm:forward-KL-discrete route after the accepted cycle-60 continuous forward-KL gate, reuse the existing cycle-51/cycle-56 discrete route, and wire main_body.tex:299-323 plus appendix.tex:260-592 through the source-cited EM/Fokker-Planck, LSI/KL/FI, DV, Gronwall, and accumulated-error interfaces without changing statements, constants, labels, or statuses.

sourceLabels : List String (explicit)

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

Ordered data items

1. thm:forward-KL-discrete
2. proof:thm:forward-KL-discrete
3. eq:frozen_interp_terminal_disc_prop_additive_final
4. lem:frozen_delta_cross_lip_sald
5. eq:KL-derivative-0-discrete
6. eq:KL-derivative-1-discrete
7. eq:KL-derivative-2-discrete
8. eq:KL-derivative-3-discrete
9. eq:frozen-cross-bound-discrete
10. eq:moving-cross-bound-discrete
11. eq:KL-derivative-5-discrete
12. eq:dv-v-term-discrete
13. eq:KL-derivative-6-discrete
14. eq:KL-derivative-7-discrete
15. lem:gronwall
16. lem:dv_variation
17. eq:LSI-KL-FI
18. thm:forward-KL

modeDiscipline : List String (explicit)

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

Ordered data items

1. Global phase judgment: cycle 60 passed reviewer/build, so no previous-cycle failure needs recovery before cycle 61 work; nevertheless the older interrupted cycle-56 discrete route is the required recovery target before any new source theorem.
2. Global phase judgment: Phase 1 theorem-skeleton translation is stable enough to continue theorem-level discrete wiring, but not stable enough for broad cited-theory, SLT, disintegration, or reusable API backfill.
3. Global phase judgment: the single lower packet that best reduces remaining proof risk is SALD.discreteForwardKlAccumulatedErrorBridgeContract / SALD.discreteForwardKlAccumulatedErrorBridgeObligation / sald.discrete_forward_kl.accumulated_error_bridge, reusing the cycle-56 pointwise Gronwall input over appendix.tex:526-592.
4. Five-backend check: lem:gronwall is represented by SALD.saldGronwallEndpointCalculusContract and theorem-specific Gronwall side conditions with endpoint-safe differentiability/FTC assumptions; lem:dv_variation is represented by dvVariationalFormulaInterface saldDvVariationSource plus finite-log-mgf/common-space witnesses; eq:LSI-KL-FI is represented by SALD.saldLsiKlFiDensityTestContract with density, zero-set, admissible-test, entropy, finite KL/FI, and Fisher-chain obligations; the continuous FP/KL derivative is represented by SALD.forwardKlDerivativeSideConditionContract and the cycle-60 lower scalar wrapper; the EM interpolation FP backend is represented by SALD.discreteForwardKlEmInterpolationSideConditionContract, endpoint laws, conditional drift, conditional-FP, and stitched-interval obligations.
5. Route order remains EM endpoint/conditional-FP -> KL derivative and frozen defect -> LSI/KL/FI -> DV velocity -> s-to-t time change -> lem:gronwall -> linear-slowdown accumulated-error bridge.
6. The six theorem skeleton consumers remain in the paper order from the analytic ledger: thm:forward-KL, thm:forward-KL-discrete, prop:guided_path_residual, thm:general-moving-target-SALD, thm:unified-forward-KL, and thm:general-moving-target-SALD-discrete.

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-discrete, change SALD.discreteForwardKlStatementContract, or promote SALD.discreteSaldContract above contractOnly.
2. Do not prove or replace EM conditional-Fokker-Planck, the frozen-defect lemma, LSI/KL/FI, source-cited DV, Gronwall, endpoint stitching, or the accumulated-error bridge in this upper packet.
3. Do not add hidden density, absolute-continuity, endpoint, finite-log-mgf, coefficient-regularity, stitched-interval, or inverse-schedule assumptions to the source theorem.
4. Do not spend this cycle on source-index rebaseline, broad SLT import, or isolated scalar sublemmas unless they are already part of the selected Gronwall/accumulated-error backend.

lowerPacket : List String (explicit)

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

Ordered data items

1. Middle should synchronize this cycle-61 upper route with conversion-windows/ASTIS-SALD-001.md and proof-obligations/ASTIS-SALD-001.md before lower work.
2. Lower should target exactly SALD.discreteForwardKlAccumulatedErrorBridgeContract / SALD.discreteForwardKlAccumulatedErrorBridgeObligation / sald.discrete_forward_kl.accumulated_error_bridge.
3. First lower sub-slice: appendix.tex:557-590 endpoint and Gronwall-output stitching, using SALD.discreteForwardKlPointwiseGronwallInputOfPostDvTimeChanged and SALD.discreteForwardKlGronwallInstantiationContract as supplied inputs.
4. Second lower sub-slice: linear slowdown t(s)=s/r and dot{s}=r coefficient collection, preserving the source constants T/(r*alpha), 2*r*eta^2*barGamma/alpha', (1/r)*A_alpha(pi,v), and 2*r*eta*barDelta_{alpha'}.
5. If proof-producing work blocks, sharpen endpoint, interval-integrability, coefficient-regularity, barGamma/barDelta, finite-log-mgf, or stitched-law obligations rather than changing theorem statements.

reviewerChecklist : List String (explicit)

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

Ordered data items

1. SALD.discreteSaldContract lists SALD.cycle61DiscreteForwardKlSkeletonObligation while remaining contractOnly.
2. SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle61_recovered_theorem_route and ASTIS.SALD.forward_KL_discrete.cycle61_lower_packet.gronwall_accumulated after the cycle-56 nodes.
3. SALD.saldDependenciesForLabel "thm:forward-KL-discrete" includes SALD.cycle61DiscreteForwardKlSkeletonUpperPacket, SALD.cycle61DiscreteForwardKlSkeletonObligation, and the cycle-61 DAG nodes.
4. The conversion window and proof-obligation ledger contain the cycle-61 global phase judgment, five-interface check, lower packet, and reviewer checklist.
5. Gronwall, DV, LSI/KL/FI, continuous FP/KL derivative, EM conditional-FP, theorem contracts, and SLT reuse statuses are not promoted.
6. 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 cycle61DiscreteForwardKlSkeletonUpperPacket :
    DiscreteForwardKlUpperPacket where
  objective := "Cycle 61 upper: recover the interrupted cycle-56 thm:forward-KL-discrete route after the accepted cycle-60 continuous forward-KL gate, reuse the existing cycle-51/cycle-56 discrete route, and wire main_body.tex:299-323 plus appendix.tex:260-592 through the source-cited EM/Fokker-Planck, LSI/KL/FI, DV, Gronwall, and accumulated-error interfaces without changing statements, constants, labels, or statuses."
  sourceLabels := [
    "thm:forward-KL-discrete",
    "proof:thm:forward-KL-discrete",
    "eq:frozen_interp_terminal_disc_prop_additive_final",
    "lem:frozen_delta_cross_lip_sald",
    "eq:KL-derivative-0-discrete",
    "eq:KL-derivative-1-discrete",
    "eq:KL-derivative-2-discrete",
    "eq:KL-derivative-3-discrete",
    "eq:frozen-cross-bound-discrete",
    "eq:moving-cross-bound-discrete",
    "eq:KL-derivative-5-discrete",
    "eq:dv-v-term-discrete",
    "eq:KL-derivative-6-discrete",
    "eq:KL-derivative-7-discrete",
    "lem:gronwall",
    "lem:dv_variation",
    "eq:LSI-KL-FI",
    "thm:forward-KL"
  ]
  modeDiscipline := [
    "Global phase judgment: cycle 60 passed reviewer/build, so no previous-cycle failure needs recovery before cycle 61 work; nevertheless the older interrupted cycle-56 discrete route is the required recovery target before any new source theorem.",
    "Global phase judgment: Phase 1 theorem-skeleton translation is stable enough to continue theorem-level discrete wiring, but not stable enough for broad cited-theory, SLT, disintegration, or reusable API backfill.",
    "Global phase judgment: the single lower packet that best reduces remaining proof risk is SALD.discreteForwardKlAccumulatedErrorBridgeContract / SALD.discreteForwardKlAccumulatedErrorBridgeObligation / sald.discrete_forward_kl.accumulated_error_bridge, reusing the cycle-56 pointwise Gronwall input over appendix.tex:526-592.",
    "Five-backend check: lem:gronwall is represented by SALD.saldGronwallEndpointCalculusContract and theorem-specific Gronwall side conditions with endpoint-safe differentiability/FTC assumptions; lem:dv_variation is represented by dvVariationalFormulaInterface saldDvVariationSource plus finite-log-mgf/common-space witnesses; eq:LSI-KL-FI is represented by SALD.saldLsiKlFiDensityTestContract with density, zero-set, admissible-test, entropy, finite KL/FI, and Fisher-chain obligations; the continuous FP/KL derivative is represented by SALD.forwardKlDerivativeSideConditionContract and the cycle-60 lower scalar wrapper; the EM interpolation FP backend is represented by SALD.discreteForwardKlEmInterpolationSideConditionContract, endpoint laws, conditional drift, conditional-FP, and stitched-interval obligations.",
    "Route order remains EM endpoint/conditional-FP -> KL derivative and frozen defect -> LSI/KL/FI -> DV velocity -> s-to-t time change -> lem:gronwall -> linear-slowdown accumulated-error bridge.",
    "The six theorem skeleton consumers remain in the paper order from the analytic ledger: thm:forward-KL, thm:forward-KL-discrete, prop:guided_path_residual, thm:general-moving-target-SALD, thm:unified-forward-KL, and thm:general-moving-target-SALD-discrete."
  ]
  nonGoals := [
    "Do not restate thm:forward-KL-discrete, change SALD.discreteForwardKlStatementContract, or promote SALD.discreteSaldContract above contractOnly.",
    "Do not prove or replace EM conditional-Fokker-Planck, the frozen-defect lemma, LSI/KL/FI, source-cited DV, Gronwall, endpoint stitching, or the accumulated-error bridge in this upper packet.",
    "Do not add hidden density, absolute-continuity, endpoint, finite-log-mgf, coefficient-regularity, stitched-interval, or inverse-schedule assumptions to the source theorem.",
    "Do not spend this cycle on source-index rebaseline, broad SLT import, or isolated scalar sublemmas unless they are already part of the selected Gronwall/accumulated-error backend."
  ]
  lowerPacket := [
    "Middle should synchronize this cycle-61 upper route with conversion-windows/ASTIS-SALD-001.md and proof-obligations/ASTIS-SALD-001.md before lower work.",
    "Lower should target exactly SALD.discreteForwardKlAccumulatedErrorBridgeContract / SALD.discreteForwardKlAccumulatedErrorBridgeObligation / sald.discrete_forward_kl.accumulated_error_bridge.",
    "First lower sub-slice: appendix.tex:557-590 endpoint and Gronwall-output stitching, using SALD.discreteForwardKlPointwiseGronwallInputOfPostDvTimeChanged and SALD.discreteForwardKlGronwallInstantiationContract as supplied inputs.",
    "Second lower sub-slice: linear slowdown t(s)=s/r and dot{s}=r coefficient collection, preserving the source constants T/(r*alpha), 2*r*eta^2*barGamma/alpha', (1/r)*A_alpha(pi,v), and 2*r*eta*barDelta_{alpha'}.",
    "If proof-producing work blocks, sharpen endpoint, interval-integrability, coefficient-regularity, barGamma/barDelta, finite-log-mgf, or stitched-law obligations rather than changing theorem statements."
  ]
  reviewerChecklist := [
    "SALD.discreteSaldContract lists SALD.cycle61DiscreteForwardKlSkeletonObligation while remaining contractOnly.",
    "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle61_recovered_theorem_route and ASTIS.SALD.forward_KL_discrete.cycle61_lower_packet.gronwall_accumulated after the cycle-56 nodes.",
    "SALD.saldDependenciesForLabel \"thm:forward-KL-discrete\" includes SALD.cycle61DiscreteForwardKlSkeletonUpperPacket, SALD.cycle61DiscreteForwardKlSkeletonObligation, and the cycle-61 DAG nodes.",
    "The conversion window and proof-obligation ledger contain the cycle-61 global phase judgment, five-interface check, lower packet, and reviewer checklist.",
    "Gronwall, DV, LSI/KL/FI, continuous FP/KL derivative, EM conditional-FP, theorem contracts, and SLT reuse statuses are not promoted.",
    "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass."
  ]
  status := ProofStatus.obligation

/-- Cycle-61 upper obligation tying the recovered discrete route to the next
Gronwall/accumulated-error lower packet. -/