AutoSamplingTheory.SALD.generalMovingTargetDvPositiveAlphaScalingObligation

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 generalMovingTargetDvPositiveAlphaScalingObligation : 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.general_moving_target.dv_positive_alpha_scaling

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 appendix lines 887-907 after DV has been applied with Z=alpha*||m_t||^2: use alpha>0 to divide alpha*E_{rho_{s(t)}}[||m_t||^2] <= K(t)+log E_{pi_t}[exp(alpha*||m_t||^2)], rewrite the log-mgf quotient as E_alpha(pi_t,m_t), and preserve the downstream coefficient sigma_t^(-2)*dot{s}(t)^(-1)*alpha^(-1) on K(t).

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldGeneralMovingTargetResidualDvSource — 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. probability.dv_variational_formula
2. def:alpha-complexity
3. SALD.saldDvFiniteLogMgfContract
4. SALD.generalMovingTargetDvFiniteLogMgfWitnessContract
5. SALD.generalMovingTargetDvPositiveAlphaScalingContract
6. KLContract

note : String (explicit)

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

This lower packet isolates only the positive-alpha scalar division and E_alpha rewrite inside the residual m_t DV witness. It adds no hypothesis to thm:general-moving-target-SALD or thm:unified-forward-KL and does not mark DV formalized.

def generalMovingTargetDvPositiveAlphaScalingObligation : ProofObligation where
  id := "sald.general_moving_target.dv_positive_alpha_scaling"
  statement := "Formalize appendix lines 887-907 after DV has been applied with Z=alpha*||m_t||^2: use alpha>0 to divide alpha*E_{rho_{s(t)}}[||m_t||^2] <= K(t)+log E_{pi_t}[exp(alpha*||m_t||^2)], rewrite the log-mgf quotient as E_alpha(pi_t,m_t), and preserve the downstream coefficient sigma_t^(-2)*dot{s}(t)^(-1)*alpha^(-1) on K(t)."
  source := saldGeneralMovingTargetResidualDvSource
  status := ProofStatus.obligation
  dependsOn := [
    "probability.dv_variational_formula",
    "def:alpha-complexity",
    "SALD.saldDvFiniteLogMgfContract",
    "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract",
    "SALD.generalMovingTargetDvPositiveAlphaScalingContract",
    "KLContract"
  ]
  note := "This lower packet isolates only the positive-alpha scalar division and E_alpha rewrite inside the residual m_t DV witness. It adds no hypothesis to thm:general-moving-target-SALD or thm:unified-forward-KL and does not mark DV formalized."