AutoSamplingTheory.SALD.cycle27DiscreteForwardKlAccumulatedCollectionLowerObligation

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.

def cycle27DiscreteForwardKlAccumulatedCollectionLowerObligation :
    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.cycle27_accumulated_collection_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.

Record the cycle-27 lower compiled scalar/integral core for thm:forward-KL-discrete: after linear slowdown supplies dot{s}(t)^(-1)=r^(-1) and dot{s}(t)=r, constant-factor extraction through the E_alpha and Delta interval integrals yields the additive collection r^(-1)*A_alpha(pi,v)+2*r*eta*barDelta_{alpha'} without changing the theorem statement.

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

1. SALD.cycle27DiscreteForwardKlMiddleContract
2. SALD.discreteForwardKlAccumulatedErrorBridgeContract
3. SALD.discreteForwardKlAlphaComplexityCollectionScalar
4. SALD.discreteForwardKlDeltaAccumulationScalar
5. SALD.discreteForwardKlAccumulatedErrorCollectionScalar
6. sald.discrete_forward_kl.accumulated_error_bridge
7. sald.discrete_forward_kl.linear_slowdown_specialization
8. def:alpha-complexity
9. sald.discrete_forward_kl.em_endpoint_laws
10. sald.discrete_forward_kl.stitched_interval_regularity
11. sald.discrete_forward_kl.residual_exponent_bound

note : String (explicit)

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

The three compiled scalar lemmas formalize only local interval-integral constant-factor algebra for alphaComplexityCollection and deltaAccumulation. Endpoint matching, A_alpha/barDelta source definitions, coefficient integrability, residual-exponent monotonicity, barGamma identification, Gronwall, DV, LSI-to-KL/FI, EM Fokker-Planck, and frozen-defect backends remain obligations.

def cycle27DiscreteForwardKlAccumulatedCollectionLowerObligation :
    ProofObligation where
  id := "sald.discrete_forward_kl.cycle27_accumulated_collection_lower"
  statement := "Record the cycle-27 lower compiled scalar/integral core for thm:forward-KL-discrete: after linear slowdown supplies dot{s}(t)^(-1)=r^(-1) and dot{s}(t)=r, constant-factor extraction through the E_alpha and Delta interval integrals yields the additive collection r^(-1)*A_alpha(pi,v)+2*r*eta*barDelta_{alpha'} without changing the theorem statement."
  source := saldForwardKlDiscreteAccumulatedErrorSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.cycle27DiscreteForwardKlMiddleContract",
    "SALD.discreteForwardKlAccumulatedErrorBridgeContract",
    "SALD.discreteForwardKlAlphaComplexityCollectionScalar",
    "SALD.discreteForwardKlDeltaAccumulationScalar",
    "SALD.discreteForwardKlAccumulatedErrorCollectionScalar",
    "sald.discrete_forward_kl.accumulated_error_bridge",
    "sald.discrete_forward_kl.linear_slowdown_specialization",
    "def:alpha-complexity",
    "sald.discrete_forward_kl.em_endpoint_laws",
    "sald.discrete_forward_kl.stitched_interval_regularity",
    "sald.discrete_forward_kl.residual_exponent_bound"
  ]
  note := "The three compiled scalar lemmas formalize only local interval-integral constant-factor algebra for alphaComplexityCollection and deltaAccumulation. Endpoint matching, A_alpha/barDelta source definitions, coefficient integrability, residual-exponent monotonicity, barGamma identification, Gronwall, DV, LSI-to-KL/FI, EM Fokker-Planck, and frozen-defect backends remain obligations."

/-- Cycle-46 upper packet for the discrete forward-KL theorem skeleton.

This keeps the cycle-44 slow analytic backend ledger in force and wires those
interfaces into `thm:forward-KL-discrete` at theorem-route level.  It is
workflow data only: the theorem remains `contractOnly`, and the EM,
Gronwall, DV, LSI/KL/FI, and KL-derivative backends remain obligations or
source-cited interfaces.
-/