AutoSamplingTheory.SALD.cycle66DiscreteForwardKlSkeletonUpperPacket

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 cycle66DiscreteForwardKlSkeletonUpperPacket :
    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 66 upper: after the accepted cycle-65 reviewer/build gate, rewire the theorem-level interfaces into thm:forward-KL-discrete, matching main_body.tex:299-323 and appendix.tex:260-592 while using the source-cited EM/Fokker-Planck, LSI/KL/FI, DV, Gronwall, and accumulated-error interfaces explicitly instead of proving them from scratch.

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. eq:discrete_delta_def
5. lem:frozen_delta_cross_lip_sald
6. eq:KL-derivative-0-discrete
7. eq:KL-derivative-1-discrete
8. eq:KL-derivative-2-discrete
9. eq:KL-derivative-3-discrete
10. eq:frozen-cross-bound-discrete
11. eq:moving-cross-bound-discrete
12. eq:KL-derivative-5-discrete
13. eq:dv-v-term-discrete
14. eq:KL-derivative-6-discrete
15. eq:KL-derivative-7-discrete
16. lem:gronwall
17. lem:dv_variation
18. eq:LSI-KL-FI
19. def:alpha-complexity
20. 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 65 passed reviewer/build, so no failed previous cycle must be recovered before cycle 66 work.
2. Global phase judgment: Phase 1 theorem-skeleton translation is stable enough for this narrow discrete forward-KL route audit, but not for broad cited-theory, SLT, SDE, disintegration, or reusable API backfill.
3. Global phase judgment: the single lower packet that best reduces the remaining discrete theorem risk is SALD.discreteForwardKlAccumulatedErrorBridgeContract / SALD.discreteForwardKlAccumulatedErrorBridgeObligation / sald.discrete_forward_kl.accumulated_error_bridge over appendix.tex:557-592 and main_body.tex:309-323.
4. Five-backend check: lem:gronwall keeps endpoint-safe differentiability/FTC, interval-integrability, coefficient-regularity, endpoint, and exponent side-condition obligations; lem:dv_variation keeps common-space, absolute-continuity, finite-KL, selected-test measurability, finite-log-mgf, and positive-alpha interfaces; eq:LSI-KL-FI keeps density, zero-set, admissible sqrt-density test, entropy identity, finite KL/FI, and Fisher chain-rule obligations; the continuous Fokker-Planck/KL derivative is visible through the cycle-65 continuous route but remains a local analytic obligation; the EM interpolation endpoint/conditional-law Fokker-Planck backend is represented by SALD.discreteForwardKlEmInterpolationSideConditionContract and its endpoint, conditional drift, conditional-FP, and stitched-interval obligations.
5. Route thm:forward-KL-discrete in the paper order: appendix.tex:260-385 EM interpolation and conditional-FP, appendix.tex:388-491 KL derivative with frozen defect and LSI, appendix.tex:493-523 DV velocity, appendix.tex:526-553 time-changed Gronwall input, and appendix.tex:557-592 Gronwall output plus accumulated-error display.
6. Preserve the exact source coefficients and constants: 4*eta^2*L_space^2<1/2, alpha in (0,alpha0), alpha' in (0,alpha0'], T/(r*alpha), 2*r*eta^2*barGamma/alpha', (1/r)*A_alpha(pi,v), and 2*r*eta*barDelta_{alpha'}.

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 the EM conditional-Fokker-Planck backend, frozen-delta lemma, LSI/KL/FI, source-cited DV, Gronwall lemma, endpoint stitching, or 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 theorem statement.
4. Do not spend this cycle on source-index rebaseline, broad SLT import, general theorem polishing, or isolated scalar sublemmas outside 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-66 route with conversion-windows/ASTIS-SALD-001.md, proof-obligations/ASTIS-SALD-001.md, research-wiki/cited-results/SLT_reuse_audit.md, and the thm:forward-KL-discrete proof DAG.
2. Lower should target exactly SALD.discreteForwardKlAccumulatedErrorBridgeContract / SALD.discreteForwardKlAccumulatedErrorBridgeObligation / sald.discrete_forward_kl.accumulated_error_bridge.
3. First lower sub-slice: appendix.tex:557-592 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, dot{s}=r, and exponent splitting, preserving the source factors exp(-r*int C_LSI), exp(T/(r*alpha)+2*r*eta^2*barGamma/alpha'), and the terminal endpoint KL(rho_K^eta||pi_T).
5. Third lower sub-slice: collect the residual integral into (1/r)*A_alpha(pi,v)+2*r*eta*barDelta_{alpha'} using the existing alpha-complexity and Delta collection scalar wrappers.
6. If blocked, sharpen endpoint-law stitching, interval-integrability, coefficient regularity, barGamma/barDelta definitions, finite-log-mgf witnesses, or residual-exponent obligations rather than changing thm:forward-KL-discrete.

reviewerChecklist : List String (explicit)

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

Ordered data items

1. SALD.discreteSaldContract lists SALD.cycle66DiscreteForwardKlSkeletonObligation while remaining contractOnly.
2. SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle66_discrete_route and ASTIS.SALD.forward_KL_discrete.cycle66_lower_packet.accumulated_error after the cycle-61 recovered route.
3. SALD.saldDependenciesForLabel "thm:forward-KL-discrete" includes SALD.cycle66DiscreteForwardKlSkeletonUpperPacket, SALD.cycle66DiscreteForwardKlSkeletonObligation, and the cycle-66 DAG nodes.
4. All five slow analytic backends remain ProofStatus.obligation or ProofStatus.sourceCited unless every analytic dependency is compiled locally.
5. No theorem statement, source coefficient, source label, source-file selection, SLT reuse entry, or analytic backend status is changed.
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 cycle66DiscreteForwardKlSkeletonUpperPacket :
    DiscreteForwardKlUpperPacket where
  objective := "Cycle 66 upper: after the accepted cycle-65 reviewer/build gate, rewire the theorem-level interfaces into thm:forward-KL-discrete, matching main_body.tex:299-323 and appendix.tex:260-592 while using the source-cited EM/Fokker-Planck, LSI/KL/FI, DV, Gronwall, and accumulated-error interfaces explicitly instead of proving them from scratch."
  sourceLabels := [
    "thm:forward-KL-discrete",
    "proof:thm:forward-KL-discrete",
    "eq:frozen_interp_terminal_disc_prop_additive_final",
    "eq:discrete_delta_def",
    "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",
    "def:alpha-complexity",
    "thm:forward-KL"
  ]
  modeDiscipline := [
    "Global phase judgment: cycle 65 passed reviewer/build, so no failed previous cycle must be recovered before cycle 66 work.",
    "Global phase judgment: Phase 1 theorem-skeleton translation is stable enough for this narrow discrete forward-KL route audit, but not for broad cited-theory, SLT, SDE, disintegration, or reusable API backfill.",
    "Global phase judgment: the single lower packet that best reduces the remaining discrete theorem risk is SALD.discreteForwardKlAccumulatedErrorBridgeContract / SALD.discreteForwardKlAccumulatedErrorBridgeObligation / sald.discrete_forward_kl.accumulated_error_bridge over appendix.tex:557-592 and main_body.tex:309-323.",
    "Five-backend check: lem:gronwall keeps endpoint-safe differentiability/FTC, interval-integrability, coefficient-regularity, endpoint, and exponent side-condition obligations; lem:dv_variation keeps common-space, absolute-continuity, finite-KL, selected-test measurability, finite-log-mgf, and positive-alpha interfaces; eq:LSI-KL-FI keeps density, zero-set, admissible sqrt-density test, entropy identity, finite KL/FI, and Fisher chain-rule obligations; the continuous Fokker-Planck/KL derivative is visible through the cycle-65 continuous route but remains a local analytic obligation; the EM interpolation endpoint/conditional-law Fokker-Planck backend is represented by SALD.discreteForwardKlEmInterpolationSideConditionContract and its endpoint, conditional drift, conditional-FP, and stitched-interval obligations.",
    "Route thm:forward-KL-discrete in the paper order: appendix.tex:260-385 EM interpolation and conditional-FP, appendix.tex:388-491 KL derivative with frozen defect and LSI, appendix.tex:493-523 DV velocity, appendix.tex:526-553 time-changed Gronwall input, and appendix.tex:557-592 Gronwall output plus accumulated-error display.",
    "Preserve the exact source coefficients and constants: 4*eta^2*L_space^2<1/2, alpha in (0,alpha0), alpha' in (0,alpha0'], T/(r*alpha), 2*r*eta^2*barGamma/alpha', (1/r)*A_alpha(pi,v), and 2*r*eta*barDelta_{alpha'}."
  ]
  nonGoals := [
    "Do not restate thm:forward-KL-discrete, change SALD.discreteForwardKlStatementContract, or promote SALD.discreteSaldContract above contractOnly.",
    "Do not prove or replace the EM conditional-Fokker-Planck backend, frozen-delta lemma, LSI/KL/FI, source-cited DV, Gronwall lemma, endpoint stitching, or 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 theorem statement.",
    "Do not spend this cycle on source-index rebaseline, broad SLT import, general theorem polishing, or isolated scalar sublemmas outside the selected Gronwall/accumulated-error backend."
  ]
  lowerPacket := [
    "Middle should synchronize this cycle-66 route with conversion-windows/ASTIS-SALD-001.md, proof-obligations/ASTIS-SALD-001.md, research-wiki/cited-results/SLT_reuse_audit.md, and the thm:forward-KL-discrete proof DAG.",
    "Lower should target exactly SALD.discreteForwardKlAccumulatedErrorBridgeContract / SALD.discreteForwardKlAccumulatedErrorBridgeObligation / sald.discrete_forward_kl.accumulated_error_bridge.",
    "First lower sub-slice: appendix.tex:557-592 endpoint and Gronwall-output stitching, using SALD.discreteForwardKlPointwiseGronwallInputOfPostDvTimeChanged and SALD.discreteForwardKlGronwallInstantiationContract as supplied inputs.",
    "Second lower sub-slice: linear slowdown t(s)=s/r, dot{s}=r, and exponent splitting, preserving the source factors exp(-r*int C_LSI), exp(T/(r*alpha)+2*r*eta^2*barGamma/alpha'), and the terminal endpoint KL(rho_K^eta||pi_T).",
    "Third lower sub-slice: collect the residual integral into (1/r)*A_alpha(pi,v)+2*r*eta*barDelta_{alpha'} using the existing alpha-complexity and Delta collection scalar wrappers.",
    "If blocked, sharpen endpoint-law stitching, interval-integrability, coefficient regularity, barGamma/barDelta definitions, finite-log-mgf witnesses, or residual-exponent obligations rather than changing thm:forward-KL-discrete."
  ]
  reviewerChecklist := [
    "SALD.discreteSaldContract lists SALD.cycle66DiscreteForwardKlSkeletonObligation while remaining contractOnly.",
    "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle66_discrete_route and ASTIS.SALD.forward_KL_discrete.cycle66_lower_packet.accumulated_error after the cycle-61 recovered route.",
    "SALD.saldDependenciesForLabel \"thm:forward-KL-discrete\" includes SALD.cycle66DiscreteForwardKlSkeletonUpperPacket, SALD.cycle66DiscreteForwardKlSkeletonObligation, and the cycle-66 DAG nodes.",
    "All five slow analytic backends remain ProofStatus.obligation or ProofStatus.sourceCited unless every analytic dependency is compiled locally.",
    "No theorem statement, source coefficient, source label, source-file selection, SLT reuse entry, or analytic backend status is changed.",
    "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass."
  ]
  status := ProofStatus.obligation

/-- Cycle-66 upper obligation tying the accepted cycle-65 continuous route back
to the focused discrete forward-KL theorem route. -/