AutoSamplingTheory.SALD.cycle173GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower1Obligation

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 cycle173GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower1Obligation :
    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_discrete.cycle173_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_hessian_source_lower1

statement : String (explicit)

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

Cycle 173 lower_1 illness-area proof-scout packet. Classification: narrows-source-cited-boundary. Exact missing theorem boundary narrowed: the live hHessianOpNorm assumption under sald.general_moving_target_discrete.em_interpolation_fp should be discharged only from a source-backed selected weak-test Hessian field, not from unexpanded testRegular, an unsourced SourceSelectedWeakTestC2bBoundedHessian predicate, or the VP score-Hessian theorem. Lower_2-ready theorem shape: selectedWeakTestHessianOpNormOfSourceHessianField (sourceTest : E -> Real) (sourceHessian : E -> E ->L[Real] (E ->L[Real] Real)) (C1 : Real) (hSourceHasHessian : forall z : E, HasFDerivAt (fderiv Real sourceTest) (sourceHessian z) z) (hSourceHessianBound : forall z : E, norm (sourceHessian z) <= C1) : forall z : E, norm (fderiv Real (fderiv Real sourceTest) z) <= C1. Proof route: intro z; use (hSourceHasHessian z).fderiv to identify fderiv Real (fderiv Real sourceTest) z with sourceHessian z; rewrite the norm; apply hSourceHessianBound z. This is a strictly smaller source-cited boundary because it separates the paper-supplied Hessian representative and its uniform bound from the downstream hHessianOpNorm consumer. It is admissible only if the source correspondence supplies hSourceHasHessian and hSourceHessianBound for the selected weak test; otherwise hHessianOpNorm remains a source-contract gap. Already compiled local support: SALD.gaussianRealStdOrthonormalBasisUnit discharges the Brownian coordinate unit direction, and SALD.gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm converts hHessianOpNorm into hSecondFDerivOpNorm. Source anchors rechecked: appendix.tex:984-995, appendix.tex:1026-1072, appendix.tex:1379-1387, main_body.tex:273-305, iteration_complexity.tex:309-321, and original SALD TeX excluding sald_version_2.tex.

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpSource — 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.cycle173GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceMiddleObligation
SALD.cycle172GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower2Obligation
SALD.gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm
SALD.gaussianRealStdOrthonormalBasisUnit
hHessianOpNorm
sourceTest
sourceHessian
HasFDerivAt.fderiv
hSourceHasHessian
hSourceHessianBound
appendix.tex:984-995
appendix.tex:1026-1072
appendix.tex:1379-1387
main_body.tex:273-305
iteration_complexity.tex:309-321 score Hessian bound rejected for sourceTest
paper-wide original-source search excluding sald_version_2.tex
source-contract-gap
sald.general_moving_target_discrete.em_interpolation_fp

note : String (explicit)

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

No local SLT file was consulted or imported. Mathlib consultation was limited to the local derivative uniqueness route via HasFDerivAt.fderiv in Mathlib.Analysis.Calculus.FDeriv.Basic; all Brownian and iterated-Frechet downstream support is already compiled locally. Lower_2 should implement this theorem only with source-backed hSourceHasHessian and hSourceHessianBound fields.

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 cycle173GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower1Obligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle173_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_hessian_source_lower1"
  statement := "Cycle 173 lower_1 illness-area proof-scout packet. Classification: narrows-source-cited-boundary. Exact missing theorem boundary narrowed: the live hHessianOpNorm assumption under sald.general_moving_target_discrete.em_interpolation_fp should be discharged only from a source-backed selected weak-test Hessian field, not from unexpanded testRegular, an unsourced SourceSelectedWeakTestC2bBoundedHessian predicate, or the VP score-Hessian theorem. Lower_2-ready theorem shape: selectedWeakTestHessianOpNormOfSourceHessianField (sourceTest : E -> Real) (sourceHessian : E -> E ->L[Real] (E ->L[Real] Real)) (C1 : Real) (hSourceHasHessian : forall z : E, HasFDerivAt (fderiv Real sourceTest) (sourceHessian z) z) (hSourceHessianBound : forall z : E, norm (sourceHessian z) <= C1) : forall z : E, norm (fderiv Real (fderiv Real sourceTest) z) <= C1. Proof route: intro z; use (hSourceHasHessian z).fderiv to identify fderiv Real (fderiv Real sourceTest) z with sourceHessian z; rewrite the norm; apply hSourceHessianBound z. This is a strictly smaller source-cited boundary because it separates the paper-supplied Hessian representative and its uniform bound from the downstream hHessianOpNorm consumer. It is admissible only if the source correspondence supplies hSourceHasHessian and hSourceHessianBound for the selected weak test; otherwise hHessianOpNorm remains a source-contract gap. Already compiled local support: SALD.gaussianRealStdOrthonormalBasisUnit discharges the Brownian coordinate unit direction, and SALD.gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm converts hHessianOpNorm into hSecondFDerivOpNorm. Source anchors rechecked: appendix.tex:984-995, appendix.tex:1026-1072, appendix.tex:1379-1387, main_body.tex:273-305, iteration_complexity.tex:309-321, and original SALD TeX excluding sald_version_2.tex."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.cycle173GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceMiddleObligation",
    "SALD.cycle172GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower2Obligation",
    "SALD.gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm",
    "SALD.gaussianRealStdOrthonormalBasisUnit",
    "hHessianOpNorm",
    "sourceTest",
    "sourceHessian",
    "HasFDerivAt.fderiv",
    "hSourceHasHessian",
    "hSourceHessianBound",
    "appendix.tex:984-995",
    "appendix.tex:1026-1072",
    "appendix.tex:1379-1387",
    "main_body.tex:273-305",
    "iteration_complexity.tex:309-321 score Hessian bound rejected for sourceTest",
    "paper-wide original-source search excluding sald_version_2.tex",
    "source-contract-gap",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "No local SLT file was consulted or imported. Mathlib consultation was limited to the local derivative uniqueness route via HasFDerivAt.fderiv in Mathlib.Analysis.Calculus.FDeriv.Basic; all Brownian and iterated-Frechet downstream support is already compiled locally. Lower_2 should implement this theorem only with source-backed hSourceHasHessian and hSourceHessianBound fields."

/-- Cycle-173 lower_2 compiled bridge from source-backed Hessian fields to the
selected-test Hessian operator-norm bound. -/

Existing module entry · Audited data-reader index · All teaching coverage