AutoSamplingTheory.SALD.cycle61DiscreteForwardKlAccumulatedErrorLowerObligation

Data definition / provenance and workflow record

Meaning and type

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

Lean statement of this data definition

The part after the colon is the output data type. This declaration takes no mathematical proof inputs.

def cycle61DiscreteForwardKlAccumulatedErrorLowerObligation :
    ProofObligation

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.

id : String (explicit)

Stable obligation identifier.

sald.discrete_forward_kl.cycle61_accumulated_error_lower

statement : String (explicit)

Desired mathematical or workflow content as String; it is not a proposition in Prop and the def does not prove it.

Cycle 61 lower compiles the scalar wrapper for the final residual-integral display in thm:forward-KL-discrete: after a separate residual-exponent argument supplies the common positive exponential factor, SALD.discreteForwardKlResidualIntegralDisplayBoundScalar uses the A_alpha and barDelta interval-collection cores to obtain exactly exp(T/(r*alpha)+2*r*eta^2*barGamma/alpha')*((1/r)*A_alpha(pi,v)+2*r*eta*barDelta_{alpha'}). EM/Fokker-Planck, endpoint stitching, residual-exponent monotonicity, barGamma/barDelta identifications, and the full accumulated-error bridge remain obligations.

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

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

status : AutoSamplingTheory.ProofStatus (explicit)

Stored ProofStatus, default obligation; even an explicitly stored formalized does not independently certify a Lean theorem.

AutoSamplingTheory.ProofStatus.obligation — stored label only; no proof certification

dependsOn : List String (explicit)

List of declared dependency names as strings; may mix theorem names, obligations, source labels, or descriptions. Not the compiled dependency DAG.

Ordered data items

SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation
SALD.discreteForwardKlAccumulatedErrorBridgeContract
SALD.discreteForwardKlAccumulatedErrorBridgeObligation
SALD.discreteForwardKlAlphaComplexityCollectionScalar
SALD.discreteForwardKlDeltaAccumulationScalar
SALD.discreteForwardKlAccumulatedErrorCollectionScalar
SALD.discreteForwardKlResidualIntegralDisplayBoundScalar
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
def:alpha-complexity

note : String (explicit)

Recorded evidence/caveats; may distinguish a compiled scalar helper from still-open source analysis.

Proof-producing scalar wrapper plus obligation only. It preserves the source constants from main_body.tex:309-323 and does not promote the theorem, Gronwall, EM/Fokker-Planck, residual exponent, endpoint stitching, or barGamma/barDelta backends.

Exact Lean data construction

Each field assignment stores the corresponding value shown above. Omitted fields use the explicitly identified schema defaults. Strings that name theorems remain strings; they do not call those theorems.

def cycle61DiscreteForwardKlAccumulatedErrorLowerObligation :
    ProofObligation where
  id := "sald.discrete_forward_kl.cycle61_accumulated_error_lower"
  statement := "Cycle 61 lower compiles the scalar wrapper for the final residual-integral display in thm:forward-KL-discrete: after a separate residual-exponent argument supplies the common positive exponential factor, SALD.discreteForwardKlResidualIntegralDisplayBoundScalar uses the A_alpha and barDelta interval-collection cores to obtain exactly exp(T/(r*alpha)+2*r*eta^2*barGamma/alpha')*((1/r)*A_alpha(pi,v)+2*r*eta*barDelta_{alpha'}). EM/Fokker-Planck, endpoint stitching, residual-exponent monotonicity, barGamma/barDelta identifications, and the full accumulated-error bridge remain obligations."
  source := saldForwardKlDiscreteAccumulatedErrorSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation",
    "SALD.discreteForwardKlAccumulatedErrorBridgeContract",
    "SALD.discreteForwardKlAccumulatedErrorBridgeObligation",
    "SALD.discreteForwardKlAlphaComplexityCollectionScalar",
    "SALD.discreteForwardKlDeltaAccumulationScalar",
    "SALD.discreteForwardKlAccumulatedErrorCollectionScalar",
    "SALD.discreteForwardKlResidualIntegralDisplayBoundScalar",
    "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",
    "def:alpha-complexity"
  ]
  note := "Proof-producing scalar wrapper plus obligation only. It preserves the source constants from main_body.tex:309-323 and does not promote the theorem, Gronwall, EM/Fokker-Planck, residual exponent, endpoint stitching, or barGamma/barDelta backends."

/-- Cycle-61 proof-DAG pane for the recovered discrete forward-KL route. -/

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