AutoSamplingTheory.SALD.cycle15DiscreteForwardKlMiddleContract

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 cycle15DiscreteForwardKlMiddleContract :
    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.

Refine the cycle-15 upper EM side-condition packet into a lower-ready source-to-Lean map for thm:forward-KL-discrete, with sald.discrete_forward_kl.em_conditional_fokker_planck as the first lower slice and all one-step and accumulated-error constants left unchanged.

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 linear slowdown t(s)=s/r and the terminal bound with Gamma, Delta, barGamma, barDelta, alpha, alpha', eta, and r.
2. appendix.tex:260-266 and 334-335 define the frozen EM interpolation hat X_s and use endpoint laws hat rho_{s_k}=rho_k^eta and hat rho_{s_{k+1}}=rho_{k+1}^eta.
3. appendix.tex:347-385 defines bar b_{k,s}, invokes partial_s hat rho_s = -div(hat rho_s*bar b_{k,s}) + Delta hat rho_s, and splits Delta hat rho_s relative to tilde pi_s.
4. appendix.tex:454-491 uses lem:frozen_delta_cross_lip_sald, Young, and LSI only after the EM conditional Fokker--Planck identity has produced the KL derivative block.
5. appendix.tex:493-523 applies the discrete DV finite-log-mgf witness for nu=hat rho_s, mu=tilde pi_s, and Z=alpha*||v_{t(s)}||^2 while preserving the dot t(s)^2 coefficient.
6. appendix.tex:526-590 changes variables from s to t, applies lem:gronwall, and leaves the main-body linear-slowdown collection to the accumulated-error bridge.

leanStepMap : List String (explicit)

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

Ordered data items

1. SALD.cycle15DiscreteForwardKlUpperPacket chooses the EM side-condition spine; this middle contract records the ordered source-to-Lean map for lower work.
2. SALD.discreteForwardKlEmInterpolationSideConditionContract splits the EM backend into sald.discrete_forward_kl.em_endpoint_laws, sald.discrete_forward_kl.em_conditional_fokker_planck, and sald.discrete_forward_kl.stitched_interval_regularity.
3. SALD.discreteForwardKlDerivativeCandidateContract depends on the conditional Fokker--Planck slice before it can use the frozen-defect lemma, LSI bridge, and DV velocity witness.
4. SALD.discreteForwardKlGronwallInstantiationContract and SALD.discreteForwardKlAccumulatedErrorBridgeContract stay separate from the EM backend so endpoint, exponent, barGamma, and barDelta algebra is not hidden.
5. SALD.discreteForwardKlCoefficientChainAuditContract remains the reviewer-facing coefficient ledger for the one-step Gamma/Delta and accumulated-error constants.

citedResultInterfaces : List String (explicit)

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

Ordered data items

1. SLT one_step_discretization remains a route reference for the EM interpolation backend only; no SLT theorem is imported or marked formalized.
2. lem:dv_variation remains source-cited through Boucheron Corollary 4.15 or the SLT entropy_duality pattern for the later velocity estimate.
3. lem:gronwall remains a local real-analysis obligation through sald.gronwall.integrating_factor and the stitched-interval interface.
4. eq:LSI-KL-FI remains the local density-test obligation probability.lsi_to_kl_fi.

obligations : List String (explicit)

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

Ordered data items

1. sald.discrete_forward_kl.cycle15_middle_em_spine
2. sald.discrete_forward_kl.conditional_drift_density
3. sald.discrete_forward_kl.em_endpoint_laws
4. sald.discrete_forward_kl.em_conditional_fokker_planck
5. sald.discrete_forward_kl.stitched_interval_regularity
6. sald.discrete_forward_kl.em_interpolation_fp
7. sald.discrete_forward_kl.frozen_delta_cross_lip
8. sald.discrete_forward_kl.kl_derivative
9. sald.discrete_forward_kl.dv_finite_log_mgf_witness
10. sald.discrete_forward_kl.dv_velocity_bound
11. sald.discrete_forward_kl.gronwall_accumulation
12. sald.discrete_forward_kl.linear_slowdown_specialization
13. sald.discrete_forward_kl.residual_exponent_bound
14. sald.discrete_forward_kl.accumulated_error_bridge
15. sald.discrete_forward_kl.coefficient_chain_audit

lowerPacket : List String (explicit)

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

Ordered data items

1. Lower target exactly one interface first: SALD.discreteForwardKlEmConditionalFpObligation, not the full theorem and not the accumulated-error bridge.
2. Use SALD.cycle15DiscreteForwardKlEmConditionalFpLowerContract as the line ledger for appendix.tex:347-385 before attempting any derivative or Gronwall proof search.
3. Inside that ledger, refine SALD.cycle15DiscreteForwardKlConditionalDriftDensityContract first: the regular conditional law, density, measurability, and integrability needed to define bar b_{k,s}.
4. For appendix.tex:347-385, expose the conditional expectation defining bar b_{k,s}, the density/law interface for hat rho_s, the frozen-interpolation Fokker--Planck equation, and the Laplacian split relative to tilde pi_s.
5. Keep endpoint law matching and stitched-interval Gronwall regularity as named sibling obligations; do not discharge the conditional Fokker--Planck slice by adding theorem-level smoothness assumptions.
6. Once the conditional Fokker--Planck backend is refined, the next lower slices remain the endpoint/stitching interfaces or the accumulated-error bridge, not a restatement of thm:forward-KL-discrete.

reviewerChecklist : List String (explicit)

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

Ordered data items

1. Every source step from appendix.tex:260-590 that is used for thm:forward-KL-discrete is classified as a Lean contract, cited result, or named proof obligation.
2. SALD.discreteSaldContract lists SALD.cycle15DiscreteForwardKlMiddleEmSpineObligation and SALD.cycle15DiscreteForwardKlConditionalDriftDensityObligation alongside the existing EM endpoint, conditional Fokker--Planck, stitched-interval, frozen-defect, DV, Gronwall, and accumulated-error obligations.
3. SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.conditional_drift_density before the conditional Fokker--Planck lower packet.
4. SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle15_conditional_fp_lower_packet before the derivative block.
5. SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle15_middle_em_spine after the upper packet and before derivative, DV, and Gronwall blocks.
6. SALD.saldDependenciesForLabel "thm:forward-KL-discrete" includes SALD.cycle15DiscreteForwardKlMiddleContract, sald.discrete_forward_kl.cycle15_middle_em_spine, and sald.discrete_forward_kl.conditional_drift_density.
7. No theorem statement, source constant, source file selection, or analytic dependency status is changed.

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 cycle15DiscreteForwardKlMiddleContract :
    DiscreteForwardKlEmDefectAccumulationMiddleContract where
  sourceStatement := saldForwardKlDiscreteSource
  interpolationSource := saldForwardKlDiscreteInterpolationSource
  derivativeSource := saldForwardKlDiscreteDerivativeSource
  gronwallSource := saldForwardKlDiscreteGronwallSource
  accumulatedSource := saldForwardKlDiscreteAccumulatedErrorSource
  objective := "Refine the cycle-15 upper EM side-condition packet into a lower-ready source-to-Lean map for thm:forward-KL-discrete, with sald.discrete_forward_kl.em_conditional_fokker_planck as the first lower slice and all one-step and accumulated-error constants left unchanged."
  sourceStepMap := [
    "main_body.tex:299-323 fixes the linear slowdown t(s)=s/r and the terminal bound with Gamma, Delta, barGamma, barDelta, alpha, alpha', eta, and r.",
    "appendix.tex:260-266 and 334-335 define the frozen EM interpolation hat X_s and use endpoint laws hat rho_{s_k}=rho_k^eta and hat rho_{s_{k+1}}=rho_{k+1}^eta.",
    "appendix.tex:347-385 defines bar b_{k,s}, invokes partial_s hat rho_s = -div(hat rho_s*bar b_{k,s}) + Delta hat rho_s, and splits Delta hat rho_s relative to tilde pi_s.",
    "appendix.tex:454-491 uses lem:frozen_delta_cross_lip_sald, Young, and LSI only after the EM conditional Fokker--Planck identity has produced the KL derivative block.",
    "appendix.tex:493-523 applies the discrete DV finite-log-mgf witness for nu=hat rho_s, mu=tilde pi_s, and Z=alpha*||v_{t(s)}||^2 while preserving the dot t(s)^2 coefficient.",
    "appendix.tex:526-590 changes variables from s to t, applies lem:gronwall, and leaves the main-body linear-slowdown collection to the accumulated-error bridge."
  ]
  leanStepMap := [
    "SALD.cycle15DiscreteForwardKlUpperPacket chooses the EM side-condition spine; this middle contract records the ordered source-to-Lean map for lower work.",
    "SALD.discreteForwardKlEmInterpolationSideConditionContract splits the EM backend into sald.discrete_forward_kl.em_endpoint_laws, sald.discrete_forward_kl.em_conditional_fokker_planck, and sald.discrete_forward_kl.stitched_interval_regularity.",
    "SALD.discreteForwardKlDerivativeCandidateContract depends on the conditional Fokker--Planck slice before it can use the frozen-defect lemma, LSI bridge, and DV velocity witness.",
    "SALD.discreteForwardKlGronwallInstantiationContract and SALD.discreteForwardKlAccumulatedErrorBridgeContract stay separate from the EM backend so endpoint, exponent, barGamma, and barDelta algebra is not hidden.",
    "SALD.discreteForwardKlCoefficientChainAuditContract remains the reviewer-facing coefficient ledger for the one-step Gamma/Delta and accumulated-error constants."
  ]
  citedResultInterfaces := [
    "SLT one_step_discretization remains a route reference for the EM interpolation backend only; no SLT theorem is imported or marked formalized.",
    "lem:dv_variation remains source-cited through Boucheron Corollary 4.15 or the SLT entropy_duality pattern for the later velocity estimate.",
    "lem:gronwall remains a local real-analysis obligation through sald.gronwall.integrating_factor and the stitched-interval interface.",
    "eq:LSI-KL-FI remains the local density-test obligation probability.lsi_to_kl_fi."
  ]
  obligations := [
    "sald.discrete_forward_kl.cycle15_middle_em_spine",
    "sald.discrete_forward_kl.conditional_drift_density",
    "sald.discrete_forward_kl.em_endpoint_laws",
    "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.frozen_delta_cross_lip",
    "sald.discrete_forward_kl.kl_derivative",
    "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",
    "sald.discrete_forward_kl.coefficient_chain_audit"
  ]
  lowerPacket := [
    "Lower target exactly one interface first: SALD.discreteForwardKlEmConditionalFpObligation, not the full theorem and not the accumulated-error bridge.",
    "Use SALD.cycle15DiscreteForwardKlEmConditionalFpLowerContract as the line ledger for appendix.tex:347-385 before attempting any derivative or Gronwall proof search.",
    "Inside that ledger, refine SALD.cycle15DiscreteForwardKlConditionalDriftDensityContract first: the regular conditional law, density, measurability, and integrability needed to define bar b_{k,s}.",
    "For appendix.tex:347-385, expose the conditional expectation defining bar b_{k,s}, the density/law interface for hat rho_s, the frozen-interpolation Fokker--Planck equation, and the Laplacian split relative to tilde pi_s.",
    "Keep endpoint law matching and stitched-interval Gronwall regularity as named sibling obligations; do not discharge the conditional Fokker--Planck slice by adding theorem-level smoothness assumptions.",
    "Once the conditional Fokker--Planck backend is refined, the next lower slices remain the endpoint/stitching interfaces or the accumulated-error bridge, not a restatement of thm:forward-KL-discrete."
  ]
  reviewerChecklist := [
    "Every source step from appendix.tex:260-590 that is used for thm:forward-KL-discrete is classified as a Lean contract, cited result, or named proof obligation.",
    "SALD.discreteSaldContract lists SALD.cycle15DiscreteForwardKlMiddleEmSpineObligation and SALD.cycle15DiscreteForwardKlConditionalDriftDensityObligation alongside the existing EM endpoint, conditional Fokker--Planck, stitched-interval, frozen-defect, DV, Gronwall, and accumulated-error obligations.",
    "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.conditional_drift_density before the conditional Fokker--Planck lower packet.",
    "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle15_conditional_fp_lower_packet before the derivative block.",
    "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle15_middle_em_spine after the upper packet and before derivative, DV, and Gronwall blocks.",
    "SALD.saldDependenciesForLabel \"thm:forward-KL-discrete\" includes SALD.cycle15DiscreteForwardKlMiddleContract, sald.discrete_forward_kl.cycle15_middle_em_spine, and sald.discrete_forward_kl.conditional_drift_density.",
    "No theorem statement, source constant, source file selection, or analytic dependency status is changed."
  ]
  status := ProofStatus.obligation