AutoSamplingTheory.SALD.saldAlphaComplexityContract

Data definition / provenance and workflow record

Meaning and type

The result has data type AutoSamplingTheory.SALD.AlphaComplexityContract. 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 saldAlphaComplexityContract : AlphaComplexityContract

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.

pathName : String (explicit): Descriptive text even when field names say formula, theorem, inequality, derivative, source gap, or proof.
pi=(pi_t)_{t in [0,T]}
velocityName : String (explicit): Descriptive text even when field names say formula, theorem, inequality, derivative, source gap, or proof.
v=(v_t)_{t in [0,T]}
alphaRange : String (explicit): Descriptive text even when field names say formula, theorem, inequality, derivative, source gap, or proof.
alpha > 0
pointwiseDensity : String (explicit): Descriptive text even when field names say formula, theorem, inequality, derivative, source gap, or proof.
E_alpha(pi_t,v_t) = alpha^{-1} * log E_{pi_t}[exp(alpha * ||v_t||^2)].
integratedComplexity : String (explicit): Descriptive text even when field names say formula, theorem, inequality, derivative, source gap, or proof.
A_alpha(pi,v) = int_0^T E_alpha(pi_t,v_t) dt.
finitenessAssumption : String (explicit): Descriptive text even when field names say formula, theorem, inequality, derivative, source gap, or proof.
thm:forward-KL assumes some alpha0 > 0 with E_{alpha0}(pi_t,v_t) < +infty for every t, then uses any alpha in (0, alpha0].
source : AutoSamplingTheory.SourceAnchor (explicit): Nested provenance data.
AutoSamplingTheory.SALD.saldAlphaComplexitySource — audited data reference, not expanded and not a compiled dependency edge
status : AutoSamplingTheory.ProofStatus (explicit): Stored workflow tag; honor the exact default but do not infer mathematical certification.
AutoSamplingTheory.ProofStatus.contractOnly — stored label only; no proof certification

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 saldAlphaComplexityContract : AlphaComplexityContract where
  pathName := "pi=(pi_t)_{t in [0,T]}"
  velocityName := "v=(v_t)_{t in [0,T]}"
  alphaRange := "alpha > 0"
  pointwiseDensity := "E_alpha(pi_t,v_t) = alpha^{-1} * log E_{pi_t}[exp(alpha * ||v_t||^2)]."
  integratedComplexity := "A_alpha(pi,v) = int_0^T E_alpha(pi_t,v_t) dt."
  finitenessAssumption := "thm:forward-KL assumes some alpha0 > 0 with E_{alpha0}(pi_t,v_t) < +infty for every t, then uses any alpha in (0, alpha0]."
  source := saldAlphaComplexitySource
  status := ProofStatus.contractOnly

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