AutoSamplingTheory.SALD.cycle180GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorMomentDag

Data definition / provenance and workflow record

Meaning and type

The result has data type List AutoSamplingTheory.ProofDagBlock. 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 cycle180GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorMomentDag :
    List ProofDagBlock

Construction and field-by-field explanation

Construct an ordered list of the following data items. It is not a logical conjunction or proof DAG.

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.

Ordered data items

id : String (explicit)
Stable plan-node identifier.
ASTIS.SALD.cycle180.middle_packet.source_hessian_gap_preserved
interface : String (explicit)
Text describing intended mathematical interface.
Preserve the cycle-179/180 source decision: hSourceHasHessian and hSourceHessianBound are not derivable from the checked original SALD source anchors and remain source-contract gaps. The Taylor moment packet does not define sourceHessian from the desired conclusion and does not use VP score-Hessian regularity.
source : AutoSamplingTheory.SourceAnchor (explicit)
Source pointer.
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpSource — audited data reference, not expanded and not a compiled dependency edge
targetLean : String (explicit)
Textual planned implementation destination.
AutoSamplingTheory/SALD.lean
dependsOn : List String (explicit)
Declared input names as strings, default empty.
Ordered data items
1. SALD.selectedWeakTestHessianOpNormOfSourceHessianField
2. hSourceHasHessian
3. hSourceHessianBound
4. appendix.tex:984-995
5. appendix.tex:1026-1072
6. appendix.tex:1170-1176
7. appendix.tex:1379-1387
8. main_body.tex:273-305
9. iteration_complexity.tex:309-321 score Hessian bound rejected for sourceTest
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. reviewer source-contract check
status : AutoSamplingTheory.ProofStatus (explicit)
Recorded workflow status, default planned.
AutoSamplingTheory.ProofStatus.obligation — stored label only; no proof certification
id : String (explicit)
Stable plan-node identifier.
ASTIS.SALD.cycle180.middle_packet.taylor_moment_integral_split
interface : String (explicit)
Text describing intended mathematical interface.
Compiled bridge: derive hFrozenScalarBrownianItoTaylorMomentDecomposition from the source coordinate-generator Taylor integral definition, integrability of the linear/quadratic/remainder summands, and the definition of remainderGeneratorLimit as the normalized-remainder integral.
source : AutoSamplingTheory.SourceAnchor (explicit)
Source pointer.
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpSource — audited data reference, not expanded and not a compiled dependency edge
targetLean : String (explicit)
Textual planned implementation destination.
AutoSamplingTheory/SALD.lean
dependsOn : List String (explicit)
Declared input names as strings, default empty.
Ordered data items
1. SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefs
2. hBrownianCoordinateGeneratorTaylorIntegralDef
3. hLinearInt
4. hQuadraticInt
5. hRemainderInt
6. hRemainderGeneratorLimitDef
7. MeasureTheory.integral_add
8. MeasureTheory.integral_const_mul
9. appendix.tex:984-995
10. appendix.tex:1170-1176
11. appendix.tex:1379-1387
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. hFrozenScalarBrownianItoTaylorMomentDecomposition
2. SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder
status : AutoSamplingTheory.ProofStatus (explicit)
Recorded workflow status, default planned.
AutoSamplingTheory.ProofStatus.formalized — stored label only; no proof certification
id : String (explicit)
Stable plan-node identifier.
ASTIS.SALD.cycle180.lower_1_packet.polynomial_summand_integrability
interface : String (explicit)
Text describing intended mathematical interface.
Compiled lower_1 bridge: discharge hLinearInt and hQuadraticInt in the Taylor moment integral split using Mathlib Gaussian exponential moments and polynomial moment integrability. The remaining Taylor-integral source boundary is hBrownianCoordinateGeneratorTaylorIntegralDef, hRemainderInt, and hRemainderGeneratorLimitDef.
source : AutoSamplingTheory.SourceAnchor (explicit)
Source pointer.
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpSource — audited data reference, not expanded and not a compiled dependency edge
targetLean : String (explicit)
Textual planned implementation destination.
AutoSamplingTheory/SALD.lean
dependsOn : List String (explicit)
Declared input names as strings, default empty.
Ordered data items
1. SALD.gaussianRealLinearQuadraticTaylorSummandsIntegrable
2. SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndGaussianPolynomialIntegrability
3. ProbabilityTheory.integrable_exp_mul_gaussianReal
4. ProbabilityTheory.integrable_pow_of_integrable_exp_mul
5. MeasureTheory.Integrable.const_mul
6. hLinearInt
7. hQuadraticInt
8. hBrownianCoordinateGeneratorTaylorIntegralDef
9. hRemainderInt
10. hRemainderGeneratorLimitDef
11. appendix.tex:984-995
12. appendix.tex:1170-1176
13. appendix.tex:1379-1387
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. hFrozenScalarBrownianItoTaylorMomentDecomposition
2. future lower_2 Taylor-integral source packet
status : AutoSamplingTheory.ProofStatus (explicit)
Recorded workflow status, default planned.
AutoSamplingTheory.ProofStatus.formalized — stored label only; no proof certification
id : String (explicit)
Stable plan-node identifier.
ASTIS.SALD.cycle180.lower_2_packet.dominated_remainder_integrability
interface : String (explicit)
Text describing intended mathematical interface.
Compiled lower_2 bridge: discharge hRemainderInt in the Taylor moment split from hRemainderMeas, hRemainderBound, and hRemainderBoundInt using MeasureTheory.Integrable.mono'. The remaining Taylor-integral source boundary is hBrownianCoordinateGeneratorTaylorIntegralDef and hRemainderGeneratorLimitDef, plus the concrete normalized-remainder measurability/domination package.
source : AutoSamplingTheory.SourceAnchor (explicit)
Source pointer.
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpSource — audited data reference, not expanded and not a compiled dependency edge
targetLean : String (explicit)
Textual planned implementation destination.
AutoSamplingTheory/SALD.lean
dependsOn : List String (explicit)
Declared input names as strings, default empty.
Ordered data items
1. SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndDominatedRemainder
2. SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndGaussianPolynomialIntegrability
3. MeasureTheory.Integrable.mono'
4. hRemainderInt
5. hRemainderMeas
6. hRemainderBound
7. hRemainderBoundInt
8. hBrownianCoordinateGeneratorTaylorIntegralDef
9. hRemainderGeneratorLimitDef
10. appendix.tex:984-995
11. appendix.tex:1170-1176
12. appendix.tex:1379-1387
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. hFrozenScalarBrownianItoTaylorMomentDecomposition
2. future source Taylor integral/remainder-limit packet
status : AutoSamplingTheory.ProofStatus (explicit)
Recorded workflow status, default planned.
AutoSamplingTheory.ProofStatus.formalized — stored label only; no proof certification
id : String (explicit)
Stable plan-node identifier.
ASTIS.SALD.cycle180.remaining_taylor_integral_source_boundary
interface : String (explicit)
Text describing intended mathematical interface.
Remaining exact source boundary after the lower_2 dominated-remainder bridge: prove hBrownianCoordinateGeneratorTaylorIntegralDef and hRemainderGeneratorLimitDef for the paper's frozen scalar Brownian Taylor expansion, and instantiate the concrete normalized-remainder measurability/domination package hRemainderMeas/hRemainderBound/hRemainderBoundInt. Keep hScalarLineTaylorCoeffDef, normalized Brownian law fields, DCT pointwise data, coordinate-sum, and Hessian source gaps separate.
source : AutoSamplingTheory.SourceAnchor (explicit)
Source pointer.
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpSource — audited data reference, not expanded and not a compiled dependency edge
targetLean : String (explicit)
Textual planned implementation destination.
AutoSamplingTheory/SALD.lean
dependsOn : List String (explicit)
Declared input names as strings, default empty.
Ordered data items
1. hBrownianCoordinateGeneratorTaylorIntegralDef
2. hRemainderGeneratorLimitDef
3. hRemainderMeas
4. hRemainderBound
5. hRemainderBoundInt
6. hScalarLineTaylorCoeffDef
7. hNormalizedVectorLaw
8. hCoordinateLawDef
9. hVarianceDef
10. hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes
11. hFrozenScalarBrownianItoEventFieldCoordinateSum
12. hSourceHasHessian
13. hSourceHessianBound
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. future lower packet
2. Brownian/Ito scalar generator bookkeeping
status : AutoSamplingTheory.ProofStatus (explicit)
Recorded workflow status, default planned.
AutoSamplingTheory.ProofStatus.obligation — 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 cycle180GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorMomentDag :
    List ProofDagBlock :=
  [
    {
      id := "ASTIS.SALD.cycle180.middle_packet.source_hessian_gap_preserved"
      interface := "Preserve the cycle-179/180 source decision: hSourceHasHessian and hSourceHessianBound are not derivable from the checked original SALD source anchors and remain source-contract gaps. The Taylor moment packet does not define sourceHessian from the desired conclusion and does not use VP score-Hessian regularity."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.selectedWeakTestHessianOpNormOfSourceHessianField",
        "hSourceHasHessian",
        "hSourceHessianBound",
        "appendix.tex:984-995",
        "appendix.tex:1026-1072",
        "appendix.tex:1170-1176",
        "appendix.tex:1379-1387",
        "main_body.tex:273-305",
        "iteration_complexity.tex:309-321 score Hessian bound rejected for sourceTest"
      ]
      reusedBy := ["reviewer source-contract check"]
      status := ProofStatus.obligation
    },
    {
      id := "ASTIS.SALD.cycle180.middle_packet.taylor_moment_integral_split"
      interface := "Compiled bridge: derive hFrozenScalarBrownianItoTaylorMomentDecomposition from the source coordinate-generator Taylor integral definition, integrability of the linear/quadratic/remainder summands, and the definition of remainderGeneratorLimit as the normalized-remainder integral."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefs",
        "hBrownianCoordinateGeneratorTaylorIntegralDef",
        "hLinearInt",
        "hQuadraticInt",
        "hRemainderInt",
        "hRemainderGeneratorLimitDef",
        "MeasureTheory.integral_add",
        "MeasureTheory.integral_const_mul",
        "appendix.tex:984-995",
        "appendix.tex:1170-1176",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "hFrozenScalarBrownianItoTaylorMomentDecomposition",
        "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle180.lower_1_packet.polynomial_summand_integrability"
      interface := "Compiled lower_1 bridge: discharge hLinearInt and hQuadraticInt in the Taylor moment integral split using Mathlib Gaussian exponential moments and polynomial moment integrability. The remaining Taylor-integral source boundary is hBrownianCoordinateGeneratorTaylorIntegralDef, hRemainderInt, and hRemainderGeneratorLimitDef."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.gaussianRealLinearQuadraticTaylorSummandsIntegrable",
        "SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndGaussianPolynomialIntegrability",
        "ProbabilityTheory.integrable_exp_mul_gaussianReal",
        "ProbabilityTheory.integrable_pow_of_integrable_exp_mul",
        "MeasureTheory.Integrable.const_mul",
        "hLinearInt",
        "hQuadraticInt",
        "hBrownianCoordinateGeneratorTaylorIntegralDef",
        "hRemainderInt",
        "hRemainderGeneratorLimitDef",
        "appendix.tex:984-995",
        "appendix.tex:1170-1176",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "hFrozenScalarBrownianItoTaylorMomentDecomposition",
        "future lower_2 Taylor-integral source packet"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle180.lower_2_packet.dominated_remainder_integrability"
      interface := "Compiled lower_2 bridge: discharge hRemainderInt in the Taylor moment split from hRemainderMeas, hRemainderBound, and hRemainderBoundInt using MeasureTheory.Integrable.mono'. The remaining Taylor-integral source boundary is hBrownianCoordinateGeneratorTaylorIntegralDef and hRemainderGeneratorLimitDef, plus the concrete normalized-remainder measurability/domination package."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndDominatedRemainder",
        "SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndGaussianPolynomialIntegrability",
        "MeasureTheory.Integrable.mono'",
        "hRemainderInt",
        "hRemainderMeas",
        "hRemainderBound",
        "hRemainderBoundInt",
        "hBrownianCoordinateGeneratorTaylorIntegralDef",
        "hRemainderGeneratorLimitDef",
        "appendix.tex:984-995",
        "appendix.tex:1170-1176",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "hFrozenScalarBrownianItoTaylorMomentDecomposition",
        "future source Taylor integral/remainder-limit packet"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle180.remaining_taylor_integral_source_boundary"
      interface := "Remaining exact source boundary after the lower_2 dominated-remainder bridge: prove hBrownianCoordinateGeneratorTaylorIntegralDef and hRemainderGeneratorLimitDef for the paper's frozen scalar Brownian Taylor expansion, and instantiate the concrete normalized-remainder measurability/domination package hRemainderMeas/hRemainderBound/hRemainderBoundInt. Keep hScalarLineTaylorCoeffDef, normalized Brownian law fields, DCT pointwise data, coordinate-sum, and Hessian source gaps separate."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "hBrownianCoordinateGeneratorTaylorIntegralDef",
        "hRemainderGeneratorLimitDef",
        "hRemainderMeas",
        "hRemainderBound",
        "hRemainderBoundInt",
        "hScalarLineTaylorCoeffDef",
        "hNormalizedVectorLaw",
        "hCoordinateLawDef",
        "hVarianceDef",
        "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes",
        "hFrozenScalarBrownianItoEventFieldCoordinateSum",
        "hSourceHasHessian",
        "hSourceHessianBound"
      ]
      reusedBy := ["future lower packet", "Brownian/Ito scalar generator bookkeeping"]
      status := ProofStatus.obligation
    }
  ]

/-! ### Cycle 183 middle: Brownian coordinate source-integral transport -/

/-- Cycle-183 middle packet for the Brownian coordinate Taylor integral leaf.

The bridge is only the `MeasureTheory.integral_congr_ae` transport from the
paper's source scalar Taylor integrand to the local Taylor-sum integrand.  It
does not prove the pointwise Taylor identity and does not revisit the selected
weak-test Hessian source-contract gap.
-/

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