AutoSamplingTheory.SALD.discreteForwardKlAccumulatedErrorBridgeObligation

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 discreteForwardKlAccumulatedErrorBridgeObligation : 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.accumulated_error_bridge

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 bridge from appendix lines 557-590 to main_body.tex lines 309-323: after Gronwall and t(s)=s/r, rewrite K(T), split exp(-int a), bound the residual exponent by the full positive exponent, and identify A_alpha(pi,v), barGamma, and barDelta_{alpha'} so the theorem has exactly the accumulated terms 2*r*eta^2*barGamma/alpha' and 2*r*eta*barDelta_{alpha'}.

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.em_endpoint_laws
2. sald.discrete_forward_kl.stitched_interval_regularity
3. sald.discrete_forward_kl.gronwall_accumulation
4. sald.discrete_forward_kl.linear_slowdown_specialization
5. sald.discrete_forward_kl.residual_exponent_bound
6. def:alpha-complexity
7. SALD.discreteForwardKlGronwallCoeffIntervalIntegrable
8. SALD.discreteForwardKlGronwallCoeffIntegralSubSub
9. SALD.discreteForwardKlGronwallInitialExponentSplitScalar
10. SALD.discreteForwardKlGronwallInitialExponentSplitOfPieces
11. SALD.discreteForwardKlAlphaComplexityCollectionScalar
12. SALD.discreteForwardKlDeltaAccumulationScalar
13. SALD.discreteForwardKlAccumulatedErrorCollectionScalar
14. SALD.discreteForwardKlResidualIntegralDisplayBoundScalar
15. SALD.discreteForwardKlMainDisplayBoundScalar
16. sald.gronwall.integrating_factor

note : String (explicit)

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

discreteForwardKlAccumulatedErrorBridgeContract isolates the final endpoint, exponent, and integral-collection algebra from the one-step derivative estimate. Cycle 46 lower compiles the three-piece initial Gronwall exponent split under explicit coefficient integrability inputs; cycle 27 lower compiles the constant-factor interval-integral core for the A_alpha and barDelta additive terms; cycle 61 lower compiles the wrapper from a supplied common-exponential residual bound to the exact main-body additive display; cycle 66 lower compiles the final scalar order wrapper from those supplied pieces to the two-term main-body display. Endpoint matching, residual-exponent monotonicity, barGamma/barDelta source identifications, and the bridge itself remain obligations, not extra theorem assumptions.

def discreteForwardKlAccumulatedErrorBridgeObligation : ProofObligation where
  id := "sald.discrete_forward_kl.accumulated_error_bridge"
  statement := "Formalize the bridge from appendix lines 557-590 to main_body.tex lines 309-323: after Gronwall and t(s)=s/r, rewrite K(T), split exp(-int a), bound the residual exponent by the full positive exponent, and identify A_alpha(pi,v), barGamma, and barDelta_{alpha'} so the theorem has exactly the accumulated terms 2*r*eta^2*barGamma/alpha' and 2*r*eta*barDelta_{alpha'}."
  source := saldForwardKlDiscreteAccumulatedErrorSource
  status := ProofStatus.obligation
  dependsOn := [
    "sald.discrete_forward_kl.em_endpoint_laws",
    "sald.discrete_forward_kl.stitched_interval_regularity",
    "sald.discrete_forward_kl.gronwall_accumulation",
    "sald.discrete_forward_kl.linear_slowdown_specialization",
    "sald.discrete_forward_kl.residual_exponent_bound",
    "def:alpha-complexity",
    "SALD.discreteForwardKlGronwallCoeffIntervalIntegrable",
    "SALD.discreteForwardKlGronwallCoeffIntegralSubSub",
    "SALD.discreteForwardKlGronwallInitialExponentSplitScalar",
    "SALD.discreteForwardKlGronwallInitialExponentSplitOfPieces",
    "SALD.discreteForwardKlAlphaComplexityCollectionScalar",
    "SALD.discreteForwardKlDeltaAccumulationScalar",
    "SALD.discreteForwardKlAccumulatedErrorCollectionScalar",
    "SALD.discreteForwardKlResidualIntegralDisplayBoundScalar",
    "SALD.discreteForwardKlMainDisplayBoundScalar",
    "sald.gronwall.integrating_factor"
  ]
  note := "discreteForwardKlAccumulatedErrorBridgeContract isolates the final endpoint, exponent, and integral-collection algebra from the one-step derivative estimate. Cycle 46 lower compiles the three-piece initial Gronwall exponent split under explicit coefficient integrability inputs; cycle 27 lower compiles the constant-factor interval-integral core for the A_alpha and barDelta additive terms; cycle 61 lower compiles the wrapper from a supplied common-exponential residual bound to the exact main-body additive display; cycle 66 lower compiles the final scalar order wrapper from those supplied pieces to the two-term main-body display. Endpoint matching, residual-exponent monotonicity, barGamma/barDelta source identifications, and the bridge itself remain obligations, not extra theorem assumptions."