AutoSamplingTheory.SALD.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoFirstOrderRemainderLowerObligation

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 cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoFirstOrderRemainderLowerObligation :
    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.cycle166_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_first_order_remainder

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 166 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hFirst input to SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff is no longer primitive once the source-facing non-quotient first-order quadratic Taylor-remainder estimate is supplied. SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder compiles the bookkeeping bridge from forall r, norm (sourceTest (x + r • e) - taylorWithinEval (fun r => sourceTest (x + r • e)) 1 Set.univ 0 r) <= C1 * r^2, plus 0 <= C1, to the exact quotient bound hFirst. Remaining source-cited work is to prove that non-quotient quadratic remainder estimate from the selected-test second-derivative/bounded-Hessian route, including Mathlib Taylor interval-to-Set.univ compatibility and the positive/negative r split. hSecondCoeff, Taylor moment decomposition, quadratic-variation normalization, and coordinate-sum leaves remain separate. Weak-FP source-field, trace-field, downstream total-event/source-functional, non-EM fallback, wrapper churn, broad audit, theorem-status promotion, fake closure, Lake/toolchain change, SLT import, and sald_version_2.tex routes are rejected.

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.formalized — 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.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder
2. SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff
3. hFirst
4. hFirstQuadraticRemainder
5. hC1
6. taylorWithinEval
7. norm_div
8. div_le_div_of_nonneg_right
9. field_simp
10. Mathlib.Analysis.Calculus.Taylor.taylor_mean_remainder_bound
11. Mathlib.Analysis.Calculus.MeanValue
12. eq:general_moving_target_SALD_frozen_interp
13. appendix.tex:984-995
14. appendix.tex:1379-1387
15. 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.

Dynamic-leaf worker packet inside the scalar Brownian/Ito illness area. Consulted local Mathlib Taylor. The available theorem taylor_mean_remainder_bound is interval-based, so the remaining named theory/source gap is the non-quotient quadratic remainder estimate over Set.univ, including interval-to-Set.univ Taylor compatibility and sign split. The configured local SLT clone is absent, and no SLT theorem was imported, ported, or marked formalized.

def cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoFirstOrderRemainderLowerObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle166_em_generator_laplacian_event_field_frozen_scalar_brownian_ito_first_order_remainder"
  statement := "Cycle 166 dynamic-leaf worker packet. Classification: narrows-source-cited-boundary. Exact boundary narrowed: the hFirst input to SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff is no longer primitive once the source-facing non-quotient first-order quadratic Taylor-remainder estimate is supplied. SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder compiles the bookkeeping bridge from forall r, norm (sourceTest (x + r • e) - taylorWithinEval (fun r => sourceTest (x + r • e)) 1 Set.univ 0 r) <= C1 * r^2, plus 0 <= C1, to the exact quotient bound hFirst. Remaining source-cited work is to prove that non-quotient quadratic remainder estimate from the selected-test second-derivative/bounded-Hessian route, including Mathlib Taylor interval-to-Set.univ compatibility and the positive/negative r split. hSecondCoeff, Taylor moment decomposition, quadratic-variation normalization, and coordinate-sum leaves remain separate. Weak-FP source-field, trace-field, downstream total-event/source-functional, non-EM fallback, wrapper churn, broad audit, theorem-status promotion, fake closure, Lake/toolchain change, SLT import, and sald_version_2.tex routes are rejected."
  source := saldGeneralMovingTargetDiscreteWeakFpSource
  status := ProofStatus.formalized
  dependsOn := [
    "SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder",
    "SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff",
    "hFirst",
    "hFirstQuadraticRemainder",
    "hC1",
    "taylorWithinEval",
    "norm_div",
    "div_le_div_of_nonneg_right",
    "field_simp",
    "Mathlib.Analysis.Calculus.Taylor.taylor_mean_remainder_bound",
    "Mathlib.Analysis.Calculus.MeanValue",
    "eq:general_moving_target_SALD_frozen_interp",
    "appendix.tex:984-995",
    "appendix.tex:1379-1387",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "Dynamic-leaf worker packet inside the scalar Brownian/Ito illness area. Consulted local Mathlib Taylor. The available theorem taylor_mean_remainder_bound is interval-based, so the remaining named theory/source gap is the non-quotient quadratic remainder estimate over Set.univ, including interval-to-Set.univ Taylor compatibility and sign split. The configured local SLT clone is absent, and no SLT theorem was imported, ported, or marked formalized."

/-- Cycle-166 lower_1 nonnegative interval Taylor proof-scout packet. -/