AutoSamplingTheory.SALD.discreteForwardKlResidualExponentBoundObligation

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 discreteForwardKlResidualExponentBoundObligation : 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.residual_exponent_bound

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 lower scalar bound used in the accumulated-error bridge: after substituting t(s)=s/r in appendix lines 557-590, for every t in [0,T] bound exp(-int_t^T a(u)du) by exp(T/(r*alpha)+2*r*eta^2*barGamma/alpha') using C_LSI(u)>=0, alpha>0, alpha'>0, r>=1, barGamma=int_0^T Gamma, and interval-integral monotonicity for the nonnegative Gamma contribution.

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.discrete_forward_kl.gronwall_accumulation
2. sald.discrete_forward_kl.linear_slowdown_specialization
3. sald.gronwall.integrating_factor
4. def:alpha-complexity
5. SALD.discreteForwardKlResidualExponentBoundScalar
6. SALD.discreteForwardKlResidualExpBoundScalar

note : String (explicit)

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

This is a lower sub-obligation of sald.discrete_forward_kl.accumulated_error_bridge. Cycle 19 lower formalized only the scalar order and Real.exp monotonicity core in SALD.discreteForwardKlResidualExponentBoundScalar and SALD.discreteForwardKlResidualExpBoundScalar; interval-integral monotonicity, coefficient positivity, and full-interval Gamma identification remain obligations. It does not change the theorem statement or the Gamma/Delta constants.

def discreteForwardKlResidualExponentBoundObligation : ProofObligation where
  id := "sald.discrete_forward_kl.residual_exponent_bound"
  statement := "Formalize the lower scalar bound used in the accumulated-error bridge: after substituting t(s)=s/r in appendix lines 557-590, for every t in [0,T] bound exp(-int_t^T a(u)du) by exp(T/(r*alpha)+2*r*eta^2*barGamma/alpha') using C_LSI(u)>=0, alpha>0, alpha'>0, r>=1, barGamma=int_0^T Gamma, and interval-integral monotonicity for the nonnegative Gamma contribution."
  source := saldForwardKlDiscreteAccumulatedErrorSource
  status := ProofStatus.obligation
  dependsOn := [
    "sald.discrete_forward_kl.gronwall_accumulation",
    "sald.discrete_forward_kl.linear_slowdown_specialization",
    "sald.gronwall.integrating_factor",
    "def:alpha-complexity",
    "SALD.discreteForwardKlResidualExponentBoundScalar",
    "SALD.discreteForwardKlResidualExpBoundScalar"
  ]
  note := "This is a lower sub-obligation of sald.discrete_forward_kl.accumulated_error_bridge. Cycle 19 lower formalized only the scalar order and Real.exp monotonicity core in SALD.discreteForwardKlResidualExponentBoundScalar and SALD.discreteForwardKlResidualExpBoundScalar; interval-integral monotonicity, coefficient positivity, and full-interval Gamma identification remain obligations. It does not change the theorem statement or the Gamma/Delta constants."