AutoSamplingTheory.SALD.cycle61DiscreteForwardKlSkeletonMiddleContract

Data definition / provenance and workflow record

Meaning and type

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

def cycle61DiscreteForwardKlSkeletonMiddleContract :
    DiscreteForwardKlEmDefectAccumulationMiddleContract

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.

sourceStatement : AutoSamplingTheory.SourceAnchor (explicit)

Nested provenance data.

AutoSamplingTheory.SALD.saldForwardKlDiscreteSource — audited data reference, not expanded and not a compiled dependency edge

interpolationSource : AutoSamplingTheory.SourceAnchor (explicit)

Nested provenance data.

AutoSamplingTheory.SALD.saldForwardKlDiscreteInterpolationSource — audited data reference, not expanded and not a compiled dependency edge

derivativeSource : AutoSamplingTheory.SourceAnchor (explicit)

Nested provenance data.

AutoSamplingTheory.SALD.saldForwardKlDiscreteDerivativeSource — audited data reference, not expanded and not a compiled dependency edge

gronwallSource : AutoSamplingTheory.SourceAnchor (explicit)

Nested provenance data.

AutoSamplingTheory.SALD.saldForwardKlDiscreteGronwallSource — audited data reference, not expanded and not a compiled dependency edge

accumulatedSource : AutoSamplingTheory.SourceAnchor (explicit)

Nested provenance data.

AutoSamplingTheory.SALD.saldForwardKlDiscreteAccumulatedErrorSource — audited data reference, not expanded and not a compiled dependency edge

objective : String (explicit)

Descriptive text even when field names say formula, theorem, inequality, derivative, source gap, or proof.

Cycle 61 middle: audit the recovered thm:forward-KL-discrete route against main_body.tex:299-323 and appendix.tex:260-592 after the accepted cycle-60 continuous route, verify that the theorem consumes source-cited EM/Fokker-Planck, LSI/KL/FI, DV, Gronwall, and accumulated-error interfaces in paper order, and select sald.discrete_forward_kl.accumulated_error_bridge over appendix.tex:557-590 plus main_body.tex:309-323 as the lower backend.

sourceStepMap : List String (explicit)

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

Ordered data items

1. main_body.tex:299-323 fixes the discrete theorem: inherited thm:forward-KL assumptions, score Lipschitz hypotheses, alpha0' finite score and 1+M complexities, 4*eta^2*L_space^2<1/2, linear slowdown t(s)=s/r, and the exact terminal constants T/(r*alpha), 2*r*eta^2*barGamma/alpha', r^(-1)*A_alpha(pi,v), and 2*r*eta*barDelta_{alpha'}.
2. appendix.tex:260-385 supplies the EM interpolation, endpoint laws, conditional drift, conditional-Fokker-Planck equation, Laplacian split, and stitched interval setup; cycle 61 continues to use these only through named obligation interfaces.
3. appendix.tex:388-491 is already routed through the cycle-51 derivative and LSI scalar handoff, including the frozen-cross quarter-FI estimate, moving Young quarter-FI estimate, and eq:LSI-KL-FI substitution.
4. appendix.tex:493-523 is already routed through the discrete DV velocity witness for nu=hat rho_s, mu=tilde pi_s, and Z=alpha*||v_{t(s)}||^2.
5. appendix.tex:526-553 is already routed through the cycle-56 pointwise t-time Gronwall input with coefficient dot{s}(t)*C_LSI(t)-dot{s}(t)^(-1)*alpha^(-1)-2*dot{s}(t)*eta^2*alpha'^(-1)*Gamma(t).
6. appendix.tex:557-590 gives the general-schedule Gronwall output, and main_body.tex:309-323 is the linear-slowdown display that must be reached by endpoint stitching, exponent splitting, barGamma/barDelta collection, and alpha-complexity collection.

leanStepMap : List String (explicit)

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

Ordered data items

1. Keep SALD.discreteForwardKlStatementContract and SALD.discreteSaldContract unchanged and contractOnly.
2. Use SALD.cycle61DiscreteForwardKlSkeletonUpperPacket, SALD.cycle61DiscreteForwardKlSkeletonObligation, SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation, SALD.cycle56DiscreteForwardKlGronwallLowerObligation, and SALD.cycle51DiscreteForwardKlDerivativeLowerObligation as parent route checks.
3. Route appendix.tex:260-385 through SALD.discreteForwardKlEmInterpolationSideConditionContract, sald.discrete_forward_kl.em_endpoint_laws, sald.discrete_forward_kl.conditional_drift_density, sald.discrete_forward_kl.em_conditional_fokker_planck, sald.discrete_forward_kl.stitched_interval_regularity, and sald.discrete_forward_kl.em_interpolation_fp.
4. Route appendix.tex:388-491 through SALD.discreteForwardKlDerivativeCandidateContract, SALD.discreteForwardKlDerivativeObligation, SALD.cycle51DiscreteForwardKlDerivativeLowerObligation, SALD.frozenDeltaCrossLipSaldContract, sald.discrete_forward_kl.frozen_delta_cross_lip, SALD.saldLsiKlFiDensityTestContract, and probability.lsi_to_kl_fi.
5. Route appendix.tex:493-523 through dvVariationalFormulaInterface saldDvVariationSource, SALD.discreteForwardKlDvFiniteLogMgfWitnessContract, sald.discrete_forward_kl.dv_finite_log_mgf_witness, and sald.discrete_forward_kl.dv_velocity_bound.
6. Route appendix.tex:526-553 through SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar, SALD.discreteForwardKlPointwiseGronwallInputOfPostDvTimeChanged, SALD.discreteForwardKlGronwallInstantiationContract, and sald.discrete_forward_kl.gronwall_accumulation.
7. Route appendix.tex:557-590 and main_body.tex:309-323 through SALD.discreteForwardKlAccumulatedErrorBridgeContract, SALD.discreteForwardKlAccumulatedErrorBridgeObligation, SALD.discreteForwardKlGronwallInitialExponentSplitOfPieces, SALD.discreteForwardKlResidualExponentBoundScalar, SALD.discreteForwardKlResidualExpBoundScalar, SALD.discreteForwardKlAlphaComplexityCollectionScalar, SALD.discreteForwardKlDeltaAccumulationScalar, SALD.discreteForwardKlAccumulatedErrorCollectionScalar, SALD.discreteForwardKlResidualIntegralDisplayBoundScalar, sald.discrete_forward_kl.linear_slowdown_specialization, sald.discrete_forward_kl.residual_exponent_bound, and sald.discrete_forward_kl.accumulated_error_bridge.
8. Select lower work on the accumulated-error bridge, not on EM conditional-FP, KL differentiation, LSI/KL/FI, DV, or the already compiled cycle-56 pointwise time-change wrapper.

citedResultInterfaces : List String (explicit)

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

Ordered data items

1. EM endpoint/conditional-law Fokker-Planck remains a source-cited obligation interface; cycle 61 middle does not prove Brownian construction, disintegration, density/AC, weak FP, boundary integration by parts, or endpoint stitching.
2. eq:LSI-KL-FI remains the density-test, zero-set, admissible-test or approximation, entropy-identity, finite KL/FI, and Fisher-chain obligation consumed by the derivative scalar handoff.
3. lem:dv_variation remains source-cited with theorem-specific common-space, absolute-continuity, finite KL, finite-log-mgf, measurability, and positive-alpha witnesses.
4. lem:gronwall remains the endpoint-safe differentiability/FTC, coefficient-regularity, interval-integrability, and exponent-rewrite obligation; the cycle-56 pointwise input is only the scalar source-shaped derivative handoff.
5. The continuous forward-KL derivative route from cycle 60 is reused only as route context for inherited assumptions and does not close any discrete EM or accumulated-error backend.

obligations : List String (explicit)

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

Ordered data items

1. sald.discrete_forward_kl.cycle61_recovered_theorem_route
2. sald.discrete_forward_kl.cycle61_middle_route_audit
3. sald.discrete_forward_kl.em_endpoint_laws
4. sald.discrete_forward_kl.conditional_drift_density
5. sald.discrete_forward_kl.em_conditional_fokker_planck
6. sald.discrete_forward_kl.stitched_interval_regularity
7. sald.discrete_forward_kl.em_interpolation_fp
8. sald.discrete_forward_kl.kl_derivative
9. sald.discrete_forward_kl.frozen_delta_cross_lip
10. probability.lsi_to_kl_fi
11. sald.discrete_forward_kl.dv_finite_log_mgf_witness
12. sald.discrete_forward_kl.dv_velocity_bound
13. sald.discrete_forward_kl.gronwall_accumulation
14. sald.discrete_forward_kl.linear_slowdown_specialization
15. sald.discrete_forward_kl.residual_exponent_bound
16. sald.discrete_forward_kl.accumulated_error_bridge

lowerPacket : List String (explicit)

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

Ordered data items

1. Target exactly SALD.discreteForwardKlAccumulatedErrorBridgeContract / SALD.discreteForwardKlAccumulatedErrorBridgeObligation / sald.discrete_forward_kl.accumulated_error_bridge.
2. First lower sub-slice: appendix.tex:557-590 endpoint and Gronwall-output stitching from the supplied cycle-56 pointwise derivative input and SALD.discreteForwardKlGronwallInstantiationContract.
3. Second lower sub-slice: linear slowdown t(s)=s/r, dot{s}=r, and exponent collection, preserving T/(r*alpha) and 2*r*eta^2*barGamma/alpha'.
4. Third lower sub-slice: collect the residual integral into r^(-1)*A_alpha(pi,v) plus 2*r*eta*barDelta_{alpha'} using SALD.discreteForwardKlAlphaComplexityCollectionScalar, SALD.discreteForwardKlDeltaAccumulationScalar, SALD.discreteForwardKlAccumulatedErrorCollectionScalar, and SALD.discreteForwardKlResidualIntegralDisplayBoundScalar.
5. If blocked, sharpen endpoint-law stitching, coefficient regularity, interval-integrability, barGamma/barDelta definitions, or finite-log-mgf 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.cycle61DiscreteForwardKlSkeletonMiddleObligation while remaining contractOnly.
2. SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle61_middle_route_audit between the recovered theorem route and the lower accumulated-error packet.
3. SALD.saldDependenciesForLabel "thm:forward-KL-discrete" includes SALD.cycle61DiscreteForwardKlSkeletonMiddleContract, SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation, and sald.discrete_forward_kl.cycle61_middle_route_audit.
4. No theorem statement, source constant, source label, theorem status, SLT reuse status, or analytic backend status is changed.
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 cycle61DiscreteForwardKlSkeletonMiddleContract :
    DiscreteForwardKlEmDefectAccumulationMiddleContract where
  sourceStatement := saldForwardKlDiscreteSource
  interpolationSource := saldForwardKlDiscreteInterpolationSource
  derivativeSource := saldForwardKlDiscreteDerivativeSource
  gronwallSource := saldForwardKlDiscreteGronwallSource
  accumulatedSource := saldForwardKlDiscreteAccumulatedErrorSource
  objective := "Cycle 61 middle: audit the recovered thm:forward-KL-discrete route against main_body.tex:299-323 and appendix.tex:260-592 after the accepted cycle-60 continuous route, verify that the theorem consumes source-cited EM/Fokker-Planck, LSI/KL/FI, DV, Gronwall, and accumulated-error interfaces in paper order, and select sald.discrete_forward_kl.accumulated_error_bridge over appendix.tex:557-590 plus main_body.tex:309-323 as the lower backend."
  sourceStepMap := [
    "main_body.tex:299-323 fixes the discrete theorem: inherited thm:forward-KL assumptions, score Lipschitz hypotheses, alpha0' finite score and 1+M complexities, 4*eta^2*L_space^2<1/2, linear slowdown t(s)=s/r, and the exact terminal constants T/(r*alpha), 2*r*eta^2*barGamma/alpha', r^(-1)*A_alpha(pi,v), and 2*r*eta*barDelta_{alpha'}.",
    "appendix.tex:260-385 supplies the EM interpolation, endpoint laws, conditional drift, conditional-Fokker-Planck equation, Laplacian split, and stitched interval setup; cycle 61 continues to use these only through named obligation interfaces.",
    "appendix.tex:388-491 is already routed through the cycle-51 derivative and LSI scalar handoff, including the frozen-cross quarter-FI estimate, moving Young quarter-FI estimate, and eq:LSI-KL-FI substitution.",
    "appendix.tex:493-523 is already routed through the discrete DV velocity witness for nu=hat rho_s, mu=tilde pi_s, and Z=alpha*||v_{t(s)}||^2.",
    "appendix.tex:526-553 is already routed through the cycle-56 pointwise t-time Gronwall input with coefficient dot{s}(t)*C_LSI(t)-dot{s}(t)^(-1)*alpha^(-1)-2*dot{s}(t)*eta^2*alpha'^(-1)*Gamma(t).",
    "appendix.tex:557-590 gives the general-schedule Gronwall output, and main_body.tex:309-323 is the linear-slowdown display that must be reached by endpoint stitching, exponent splitting, barGamma/barDelta collection, and alpha-complexity collection."
  ]
  leanStepMap := [
    "Keep SALD.discreteForwardKlStatementContract and SALD.discreteSaldContract unchanged and contractOnly.",
    "Use SALD.cycle61DiscreteForwardKlSkeletonUpperPacket, SALD.cycle61DiscreteForwardKlSkeletonObligation, SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation, SALD.cycle56DiscreteForwardKlGronwallLowerObligation, and SALD.cycle51DiscreteForwardKlDerivativeLowerObligation as parent route checks.",
    "Route appendix.tex:260-385 through SALD.discreteForwardKlEmInterpolationSideConditionContract, sald.discrete_forward_kl.em_endpoint_laws, sald.discrete_forward_kl.conditional_drift_density, sald.discrete_forward_kl.em_conditional_fokker_planck, sald.discrete_forward_kl.stitched_interval_regularity, and sald.discrete_forward_kl.em_interpolation_fp.",
    "Route appendix.tex:388-491 through SALD.discreteForwardKlDerivativeCandidateContract, SALD.discreteForwardKlDerivativeObligation, SALD.cycle51DiscreteForwardKlDerivativeLowerObligation, SALD.frozenDeltaCrossLipSaldContract, sald.discrete_forward_kl.frozen_delta_cross_lip, SALD.saldLsiKlFiDensityTestContract, and probability.lsi_to_kl_fi.",
    "Route appendix.tex:493-523 through dvVariationalFormulaInterface saldDvVariationSource, SALD.discreteForwardKlDvFiniteLogMgfWitnessContract, sald.discrete_forward_kl.dv_finite_log_mgf_witness, and sald.discrete_forward_kl.dv_velocity_bound.",
    "Route appendix.tex:526-553 through SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar, SALD.discreteForwardKlPointwiseGronwallInputOfPostDvTimeChanged, SALD.discreteForwardKlGronwallInstantiationContract, and sald.discrete_forward_kl.gronwall_accumulation.",
    "Route appendix.tex:557-590 and main_body.tex:309-323 through SALD.discreteForwardKlAccumulatedErrorBridgeContract, SALD.discreteForwardKlAccumulatedErrorBridgeObligation, SALD.discreteForwardKlGronwallInitialExponentSplitOfPieces, SALD.discreteForwardKlResidualExponentBoundScalar, SALD.discreteForwardKlResidualExpBoundScalar, SALD.discreteForwardKlAlphaComplexityCollectionScalar, SALD.discreteForwardKlDeltaAccumulationScalar, SALD.discreteForwardKlAccumulatedErrorCollectionScalar, SALD.discreteForwardKlResidualIntegralDisplayBoundScalar, sald.discrete_forward_kl.linear_slowdown_specialization, sald.discrete_forward_kl.residual_exponent_bound, and sald.discrete_forward_kl.accumulated_error_bridge.",
    "Select lower work on the accumulated-error bridge, not on EM conditional-FP, KL differentiation, LSI/KL/FI, DV, or the already compiled cycle-56 pointwise time-change wrapper."
  ]
  citedResultInterfaces := [
    "EM endpoint/conditional-law Fokker-Planck remains a source-cited obligation interface; cycle 61 middle does not prove Brownian construction, disintegration, density/AC, weak FP, boundary integration by parts, or endpoint stitching.",
    "eq:LSI-KL-FI remains the density-test, zero-set, admissible-test or approximation, entropy-identity, finite KL/FI, and Fisher-chain obligation consumed by the derivative scalar handoff.",
    "lem:dv_variation remains source-cited with theorem-specific common-space, absolute-continuity, finite KL, finite-log-mgf, measurability, and positive-alpha witnesses.",
    "lem:gronwall remains the endpoint-safe differentiability/FTC, coefficient-regularity, interval-integrability, and exponent-rewrite obligation; the cycle-56 pointwise input is only the scalar source-shaped derivative handoff.",
    "The continuous forward-KL derivative route from cycle 60 is reused only as route context for inherited assumptions and does not close any discrete EM or accumulated-error backend."
  ]
  obligations := [
    "sald.discrete_forward_kl.cycle61_recovered_theorem_route",
    "sald.discrete_forward_kl.cycle61_middle_route_audit",
    "sald.discrete_forward_kl.em_endpoint_laws",
    "sald.discrete_forward_kl.conditional_drift_density",
    "sald.discrete_forward_kl.em_conditional_fokker_planck",
    "sald.discrete_forward_kl.stitched_interval_regularity",
    "sald.discrete_forward_kl.em_interpolation_fp",
    "sald.discrete_forward_kl.kl_derivative",
    "sald.discrete_forward_kl.frozen_delta_cross_lip",
    "probability.lsi_to_kl_fi",
    "sald.discrete_forward_kl.dv_finite_log_mgf_witness",
    "sald.discrete_forward_kl.dv_velocity_bound",
    "sald.discrete_forward_kl.gronwall_accumulation",
    "sald.discrete_forward_kl.linear_slowdown_specialization",
    "sald.discrete_forward_kl.residual_exponent_bound",
    "sald.discrete_forward_kl.accumulated_error_bridge"
  ]
  lowerPacket := [
    "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 from the supplied cycle-56 pointwise derivative input and SALD.discreteForwardKlGronwallInstantiationContract.",
    "Second lower sub-slice: linear slowdown t(s)=s/r, dot{s}=r, and exponent collection, preserving T/(r*alpha) and 2*r*eta^2*barGamma/alpha'.",
    "Third lower sub-slice: collect the residual integral into r^(-1)*A_alpha(pi,v) plus 2*r*eta*barDelta_{alpha'} using SALD.discreteForwardKlAlphaComplexityCollectionScalar, SALD.discreteForwardKlDeltaAccumulationScalar, SALD.discreteForwardKlAccumulatedErrorCollectionScalar, and SALD.discreteForwardKlResidualIntegralDisplayBoundScalar.",
    "If blocked, sharpen endpoint-law stitching, coefficient regularity, interval-integrability, barGamma/barDelta definitions, or finite-log-mgf obligations rather than changing thm:forward-KL-discrete."
  ]
  reviewerChecklist := [
    "SALD.discreteSaldContract lists SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation while remaining contractOnly.",
    "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle61_middle_route_audit between the recovered theorem route and the lower accumulated-error packet.",
    "SALD.saldDependenciesForLabel \"thm:forward-KL-discrete\" includes SALD.cycle61DiscreteForwardKlSkeletonMiddleContract, SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation, and sald.discrete_forward_kl.cycle61_middle_route_audit.",
    "No theorem statement, source constant, source label, theorem status, SLT reuse status, 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-61 middle obligation tying the recovered theorem route to the
accumulated-error bridge lower packet. -/