AutoSamplingTheory.SALD.forwardKlDvFiniteLogMgfWitnessObligation

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 forwardKlDvFiniteLogMgfWitnessObligation : 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.forward_kl.dv_finite_log_mgf_witness

statement : String (explicit)

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

Formalize the side interface for appendix lines 230-241: with nu=rho_{s(t)}, mu=pi_t, and Z=alpha*||v_t||^2, derive the finite log-mgf condition for alpha in (0,alpha0] from E_{alpha0}(pi_t,v_t)<+infty, keep rho_{s(t)} and pi_t on the common state space with the needed absolute continuity, and divide the DV inequality by alpha>0 without changing the source coefficient.

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldForwardKlDvEnergySource — 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.cycle26ForwardKlUpperPacket
SALD.cycle26ForwardKlMiddleContract
sald.forward_kl.cycle26_dv_witness_middle
probability.dv_variational_formula
def:alpha-complexity
SALD.saldDvFiniteLogMgfContract
sald.dv_variation.finite_log_mgf_interface
sald.forward_kl.dv_alpha_mgf_monotonicity
SALD.forwardKlDvPositiveAlphaScalingScalar
SALD.forwardKlDvPositiveAlphaCoefficientScalar
sald.forward_kl.moving_target_dependency_chain

note : String (explicit)

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

forwardKlDvFiniteLogMgfWitnessContract narrows the Cycle 10 middle target to the theorem-specific DV witness. Cycle 26 lower now compiles only the positive-alpha scalar division and coefficient preservation; DV, common-space, measurability, and exponential-moment monotonicity remain obligations.

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 forwardKlDvFiniteLogMgfWitnessObligation : ProofObligation where
  id := "sald.forward_kl.dv_finite_log_mgf_witness"
  statement := "Formalize the side interface for appendix lines 230-241: with nu=rho_{s(t)}, mu=pi_t, and Z=alpha*||v_t||^2, derive the finite log-mgf condition for alpha in (0,alpha0] from E_{alpha0}(pi_t,v_t)<+infty, keep rho_{s(t)} and pi_t on the common state space with the needed absolute continuity, and divide the DV inequality by alpha>0 without changing the source coefficient."
  source := saldForwardKlDvEnergySource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.cycle26ForwardKlUpperPacket",
    "SALD.cycle26ForwardKlMiddleContract",
    "sald.forward_kl.cycle26_dv_witness_middle",
    "probability.dv_variational_formula",
    "def:alpha-complexity",
    "SALD.saldDvFiniteLogMgfContract",
    "sald.dv_variation.finite_log_mgf_interface",
    "sald.forward_kl.dv_alpha_mgf_monotonicity",
    "SALD.forwardKlDvPositiveAlphaScalingScalar",
    "SALD.forwardKlDvPositiveAlphaCoefficientScalar",
    "sald.forward_kl.moving_target_dependency_chain"
  ]
  note := "forwardKlDvFiniteLogMgfWitnessContract narrows the Cycle 10 middle target to the theorem-specific DV witness. Cycle 26 lower now compiles only the positive-alpha scalar division and coefficient preservation; DV, common-space, measurability, and exponential-moment monotonicity remain obligations."

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