AutoSamplingTheory.SALD.cycle27DiscreteForwardKlMiddleContract

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 cycle27DiscreteForwardKlMiddleContract :
    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.

Translate the cycle-27 accumulated-collection target into a lower-ready map for sald.discrete_forward_kl.accumulated_error_bridge, with endpointBridge plus alphaComplexityCollection and deltaAccumulation as the first sub-slice and all source constants preserved.

sourceStepMap : List String (explicit)

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

Ordered data items

1. appendix.tex:557-590: start from the general-schedule Gronwall output for K(T), with a(t)=dot{s}(t)*C_LSI(t)-dot{s}(t)^(-1)*alpha^(-1)-2*dot{s}(t)*eta^2*alpha'^(-1)*Gamma(t).
2. appendix.tex:560-571: the endpoint term must rewrite K(T) to KL(rho_K^eta||pi_T) and K(0) to KL(rho_0||pi_0) using EM endpoint laws and linear-slowdown endpoints.
3. appendix.tex:586-589: the residual integrand is dot{s}(t)^(-1)*E_alpha(pi_t,v_t)+2*dot{s}(t)*eta*Delta(t).
4. main_body.tex:299-323: under t(s)=s/r, dot{s}(t)=r and dot{s}(t)^(-1)=1/r, so the additive residual collection is (1/r)*A_alpha(pi,v)+2*r*eta*barDelta_{alpha'}.
5. main_body.tex:310-323: the positive exponent still uses T/(r*alpha)+2*r*eta^2*barGamma/alpha'; residual-exponent monotonicity and barGamma identification remain separate obligations.

leanStepMap : List String (explicit)

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

Ordered data items

1. The theorem statement remains SALD.discreteForwardKlStatementContract and SALD.discreteSaldContract; this packet adds no hypothesis and proves no terminal KL bound.
2. The lower-facing target is SALD.discreteForwardKlAccumulatedErrorBridgeContract / SALD.discreteForwardKlAccumulatedErrorBridgeObligation / sald.discrete_forward_kl.accumulated_error_bridge.
3. The first lower sub-slice is the contract fields endpointBridge, alphaComplexityCollection, and deltaAccumulation.
4. Endpoint rewrites depend on sald.discrete_forward_kl.em_endpoint_laws, sald.discrete_forward_kl.stitched_interval_regularity, and sald.discrete_forward_kl.linear_slowdown_specialization.
5. The A_alpha and barDelta collections use def:alpha-complexity and the main-body definitions of barGamma and barDelta; no new constants are introduced.
6. SALD.discreteForwardKlResidualExponentBoundScalar and SALD.discreteForwardKlResidualExpBoundScalar remain only scalar cores for the residual exponent dependency, not a completed accumulated-error bridge.

citedResultInterfaces : List String (explicit)

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

Ordered data items

1. lem:gronwall remains a local real-analysis obligation through sald.discrete_forward_kl.gronwall_accumulation and sald.gronwall.integrating_factor.
2. lem:dv_variation remains inherited through the existing discrete DV velocity witness and is not reopened by this packet.
3. eq:LSI-KL-FI remains inherited only through the nonnegative LSI contribution used by the residual exponent bound.
4. No SLT theorem applies to endpoint rewrites, alpha-complexity collection, Delta accumulation, barGamma identification, or interval-integral monotonicity.

obligations : List String (explicit)

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

Ordered data items

1. sald.discrete_forward_kl.cycle27_accumulated_collection_middle
2. sald.discrete_forward_kl.cycle27_accumulated_collection_lower
3. sald.discrete_forward_kl.accumulated_error_bridge
4. sald.discrete_forward_kl.linear_slowdown_specialization
5. sald.discrete_forward_kl.residual_exponent_bound
6. sald.discrete_forward_kl.gronwall_accumulation
7. sald.discrete_forward_kl.em_endpoint_laws
8. sald.discrete_forward_kl.stitched_interval_regularity
9. sald.discrete_forward_kl.coefficient_chain_audit
10. def:alpha-complexity
11. SALD.discreteForwardKlAlphaComplexityCollectionScalar
12. SALD.discreteForwardKlDeltaAccumulationScalar
13. SALD.discreteForwardKlAccumulatedErrorCollectionScalar
14. sald.gronwall.integrating_factor

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 sub-slice: endpointBridge plus alphaComplexityCollection and deltaAccumulation, using appendix.tex:557-590 and main_body.tex:309-323.
3. Instantiate dot{s}=r and dot{s}^(-1)=1/r only through the existing linear-slowdown obligation; do not add those identities as theorem hypotheses.
4. Keep sald.discrete_forward_kl.residual_exponent_bound and the barGamma full-interval identification as dependencies unless the lower slice explicitly proves them.
5. Do not reopen the frozen-defect, LSI, DV, time-change coefficient, or full Gronwall subproofs.

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.cycle27_middle_accumulated_collection between the cycle-27 upper packet and the accumulated-error bridge.
2. SALD.saldDependenciesForLabel "thm:forward-KL-discrete" includes SALD.cycle27DiscreteForwardKlMiddleContract and sald.discrete_forward_kl.cycle27_accumulated_collection_middle.
3. The conversion window, proof-obligation ledger, and SLT audit classify this packet as workflow/source-to-Lean data, not as a proof of thm:forward-KL-discrete.
4. The constants T/(r*alpha), 2*r*eta^2*barGamma/alpha', (1/r)*A_alpha(pi,v), and 2*r*eta*barDelta_{alpha'} are unchanged.
5. No analytic dependency is promoted beyond obligation/source-cited status.

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 cycle27DiscreteForwardKlMiddleContract :
    DiscreteForwardKlEmDefectAccumulationMiddleContract where
  sourceStatement := saldForwardKlDiscreteSource
  interpolationSource := saldForwardKlDiscreteInterpolationSource
  derivativeSource := saldForwardKlDiscreteDerivativeSource
  gronwallSource := saldForwardKlDiscreteGronwallSource
  accumulatedSource := saldForwardKlDiscreteAccumulatedErrorSource
  objective := "Translate the cycle-27 accumulated-collection target into a lower-ready map for sald.discrete_forward_kl.accumulated_error_bridge, with endpointBridge plus alphaComplexityCollection and deltaAccumulation as the first sub-slice and all source constants preserved."
  sourceStepMap := [
    "appendix.tex:557-590: start from the general-schedule Gronwall output for K(T), with a(t)=dot{s}(t)*C_LSI(t)-dot{s}(t)^(-1)*alpha^(-1)-2*dot{s}(t)*eta^2*alpha'^(-1)*Gamma(t).",
    "appendix.tex:560-571: the endpoint term must rewrite K(T) to KL(rho_K^eta||pi_T) and K(0) to KL(rho_0||pi_0) using EM endpoint laws and linear-slowdown endpoints.",
    "appendix.tex:586-589: the residual integrand is dot{s}(t)^(-1)*E_alpha(pi_t,v_t)+2*dot{s}(t)*eta*Delta(t).",
    "main_body.tex:299-323: under t(s)=s/r, dot{s}(t)=r and dot{s}(t)^(-1)=1/r, so the additive residual collection is (1/r)*A_alpha(pi,v)+2*r*eta*barDelta_{alpha'}.",
    "main_body.tex:310-323: the positive exponent still uses T/(r*alpha)+2*r*eta^2*barGamma/alpha'; residual-exponent monotonicity and barGamma identification remain separate obligations."
  ]
  leanStepMap := [
    "The theorem statement remains SALD.discreteForwardKlStatementContract and SALD.discreteSaldContract; this packet adds no hypothesis and proves no terminal KL bound.",
    "The lower-facing target is SALD.discreteForwardKlAccumulatedErrorBridgeContract / SALD.discreteForwardKlAccumulatedErrorBridgeObligation / sald.discrete_forward_kl.accumulated_error_bridge.",
    "The first lower sub-slice is the contract fields endpointBridge, alphaComplexityCollection, and deltaAccumulation.",
    "Endpoint rewrites depend on sald.discrete_forward_kl.em_endpoint_laws, sald.discrete_forward_kl.stitched_interval_regularity, and sald.discrete_forward_kl.linear_slowdown_specialization.",
    "The A_alpha and barDelta collections use def:alpha-complexity and the main-body definitions of barGamma and barDelta; no new constants are introduced.",
    "SALD.discreteForwardKlResidualExponentBoundScalar and SALD.discreteForwardKlResidualExpBoundScalar remain only scalar cores for the residual exponent dependency, not a completed accumulated-error bridge."
  ]
  citedResultInterfaces := [
    "lem:gronwall remains a local real-analysis obligation through sald.discrete_forward_kl.gronwall_accumulation and sald.gronwall.integrating_factor.",
    "lem:dv_variation remains inherited through the existing discrete DV velocity witness and is not reopened by this packet.",
    "eq:LSI-KL-FI remains inherited only through the nonnegative LSI contribution used by the residual exponent bound.",
    "No SLT theorem applies to endpoint rewrites, alpha-complexity collection, Delta accumulation, barGamma identification, or interval-integral monotonicity."
  ]
  obligations := [
      "sald.discrete_forward_kl.cycle27_accumulated_collection_middle",
      "sald.discrete_forward_kl.cycle27_accumulated_collection_lower",
      "sald.discrete_forward_kl.accumulated_error_bridge",
      "sald.discrete_forward_kl.linear_slowdown_specialization",
      "sald.discrete_forward_kl.residual_exponent_bound",
    "sald.discrete_forward_kl.gronwall_accumulation",
    "sald.discrete_forward_kl.em_endpoint_laws",
      "sald.discrete_forward_kl.stitched_interval_regularity",
      "sald.discrete_forward_kl.coefficient_chain_audit",
      "def:alpha-complexity",
      "SALD.discreteForwardKlAlphaComplexityCollectionScalar",
      "SALD.discreteForwardKlDeltaAccumulationScalar",
      "SALD.discreteForwardKlAccumulatedErrorCollectionScalar",
      "sald.gronwall.integrating_factor"
    ]
  lowerPacket := [
    "Target exactly SALD.discreteForwardKlAccumulatedErrorBridgeContract / SALD.discreteForwardKlAccumulatedErrorBridgeObligation / sald.discrete_forward_kl.accumulated_error_bridge.",
    "First sub-slice: endpointBridge plus alphaComplexityCollection and deltaAccumulation, using appendix.tex:557-590 and main_body.tex:309-323.",
    "Instantiate dot{s}=r and dot{s}^(-1)=1/r only through the existing linear-slowdown obligation; do not add those identities as theorem hypotheses.",
    "Keep sald.discrete_forward_kl.residual_exponent_bound and the barGamma full-interval identification as dependencies unless the lower slice explicitly proves them.",
    "Do not reopen the frozen-defect, LSI, DV, time-change coefficient, or full Gronwall subproofs."
  ]
  reviewerChecklist := [
    "SALD.discreteForwardKlProofDag contains ASTIS.SALD.forward_KL_discrete.cycle27_middle_accumulated_collection between the cycle-27 upper packet and the accumulated-error bridge.",
    "SALD.saldDependenciesForLabel \"thm:forward-KL-discrete\" includes SALD.cycle27DiscreteForwardKlMiddleContract and sald.discrete_forward_kl.cycle27_accumulated_collection_middle.",
    "The conversion window, proof-obligation ledger, and SLT audit classify this packet as workflow/source-to-Lean data, not as a proof of thm:forward-KL-discrete.",
    "The constants T/(r*alpha), 2*r*eta^2*barGamma/alpha', (1/r)*A_alpha(pi,v), and 2*r*eta*barDelta_{alpha'} are unchanged.",
    "No analytic dependency is promoted beyond obligation/source-cited status."
  ]
  status := ProofStatus.obligation