AutoSamplingTheory.SALD.cycle184GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralLawDag

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 cycle184GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralLawDag :
    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.cycle184.middle_packet.source_integral_law_bridge
interface : String (explicit)
Text describing intended mathematical interface.
Compiled bridge: derive hBrownianCoordinateGeneratorSourceIntegralDef from hBrownianCoordinateGeneratorNormalizedLawDef, hNormalizedVectorLaw, hCoordinateLawDef, and hVarianceDef by reusing SALD.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw and SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw.
source : AutoSamplingTheory.SourceAnchor (explicit)
Source pointer.
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteEmSource — 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.selectedWeakTestBrownianCoordinateGeneratorSourceIntegralDefOfStdGaussianVectorLaw
2. SALD.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw
3. SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw
4. hBrownianCoordinateGeneratorNormalizedLawDef
5. hNormalizedVectorLaw
6. hCoordinateLawDef
7. hVarianceDef
8. appendix.tex:958-996
9. appendix.tex:1170-1176
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. hBrownianCoordinateGeneratorSourceIntegralDef
2. SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralAndAE
3. SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndDominatedRemainder
4. sald.general_moving_target_discrete.em_interpolation_fp
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.cycle184.lower_1_packet.normalized_law_route
interface : String (explicit)
Text describing intended mathematical interface.
Natural-language route for the remaining source law definition: prove hBrownianCoordinateGeneratorNormalizedLawDef by defining brownianCoordinateGenerator as the expectation of the paper sourceTaylorIntegrand over the normalized scalar Brownian coordinate induced by eq:general_moving_target_SALD_frozen_interp, before replacing that law by gaussianReal. Keep the scalar Taylor identity and DCT remainder package separate.
source : AutoSamplingTheory.SourceAnchor (explicit)
Source pointer.
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteEmSource — 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. hBrownianCoordinateGeneratorNormalizedLawDef
2. sourceTaylorIntegrand
3. normalizedCoordinateLaw
4. eq:general_moving_target_SALD_frozen_interp
5. appendix.tex:984-995
6. appendix.tex:1170-1176
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. SALD.selectedWeakTestBrownianCoordinateGeneratorNormalizedLawDefOfScalarPushforward
2. hBrownianCoordinateGeneratorNormalizedLawDef
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.cycle184.lower_2_packet.normalized_law_scalar_pushforward
interface : String (explicit)
Text describing intended mathematical interface.
Compiled lower_2 bridge: derive hBrownianCoordinateGeneratorNormalizedLawDef from the scalar Brownian coordinate measurability hScalarMeas, the normalized law pushforward definition hNormalizedCoordinateLawDef, source integrand measurability hSourceTaylorIntegrandMeas, and the sample-space generator definition hGeneratorPullbackDef by MeasureTheory.integral_map.
source : AutoSamplingTheory.SourceAnchor (explicit)
Source pointer.
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteEmSource — 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.selectedWeakTestBrownianCoordinateGeneratorNormalizedLawDefOfScalarPushforward
2. MeasureTheory.integral_map
3. hScalarMeas
4. hNormalizedCoordinateLawDef
5. hSourceTaylorIntegrandMeas
6. hGeneratorPullbackDef
7. scalarBrownianCoordinate
8. sourceTaylorIntegrand
9. normalizedCoordinateLaw
10. appendix.tex:984-995
11. appendix.tex:1170-1176
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. hBrownianCoordinateGeneratorNormalizedLawDef
2. SALD.selectedWeakTestBrownianCoordinateGeneratorSourceIntegralDefOfStdGaussianVectorLaw
3. SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralAndAE
4. sald.general_moving_target_discrete.em_interpolation_fp
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.cycle184.remaining_brownian_taylor_backend
interface : String (explicit)
Text describing intended mathematical interface.
Remaining exact Brownian/Ito frozen backend after the lower_2 scalar-pushforward bridge: hScalarMeas, hNormalizedCoordinateLawDef, hSourceTaylorIntegrandMeas, hGeneratorPullbackDef, hSourceTaylorIntegrandPointwise, hRemainderGeneratorLimitDef, and hRemainderMeas/hRemainderBound/hRemainderBoundInt. hSourceHasHessian and hSourceHessianBound stay source-contract gaps.
source : AutoSamplingTheory.SourceAnchor (explicit)
Source pointer.
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteEmSource — 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. hScalarMeas
2. hNormalizedCoordinateLawDef
3. hSourceTaylorIntegrandMeas
4. hGeneratorPullbackDef
5. hSourceTaylorIntegrandPointwise
6. hRemainderGeneratorLimitDef
7. hRemainderMeas
8. hRemainderBound
9. hRemainderBoundInt
10. hSourceHasHessian
11. hSourceHessianBound
12. source-contract-gap for selected weak-test Hessian fields
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. sald.general_moving_target_discrete.em_interpolation_fp
2. sald.general_moving_target_discrete.kl_derivative
3. divergence/FI/IBP handoff
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 cycle184GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralLawDag :
    List ProofDagBlock :=
  [
    {
      id := "ASTIS.SALD.cycle184.middle_packet.source_integral_law_bridge"
      interface := "Compiled bridge: derive hBrownianCoordinateGeneratorSourceIntegralDef from hBrownianCoordinateGeneratorNormalizedLawDef, hNormalizedVectorLaw, hCoordinateLawDef, and hVarianceDef by reusing SALD.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw and SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw."
      source := saldGeneralMovingTargetDiscreteEmSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.selectedWeakTestBrownianCoordinateGeneratorSourceIntegralDefOfStdGaussianVectorLaw",
        "SALD.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw",
        "SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw",
        "hBrownianCoordinateGeneratorNormalizedLawDef",
        "hNormalizedVectorLaw",
        "hCoordinateLawDef",
        "hVarianceDef",
        "appendix.tex:958-996",
        "appendix.tex:1170-1176"
      ]
      reusedBy := [
        "hBrownianCoordinateGeneratorSourceIntegralDef",
        "SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralAndAE",
        "SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndDominatedRemainder",
        "sald.general_moving_target_discrete.em_interpolation_fp"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle184.lower_1_packet.normalized_law_route"
      interface := "Natural-language route for the remaining source law definition: prove hBrownianCoordinateGeneratorNormalizedLawDef by defining brownianCoordinateGenerator as the expectation of the paper sourceTaylorIntegrand over the normalized scalar Brownian coordinate induced by eq:general_moving_target_SALD_frozen_interp, before replacing that law by gaussianReal. Keep the scalar Taylor identity and DCT remainder package separate."
      source := saldGeneralMovingTargetDiscreteEmSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "hBrownianCoordinateGeneratorNormalizedLawDef",
        "sourceTaylorIntegrand",
        "normalizedCoordinateLaw",
        "eq:general_moving_target_SALD_frozen_interp",
        "appendix.tex:984-995",
        "appendix.tex:1170-1176"
      ]
      reusedBy := [
        "SALD.selectedWeakTestBrownianCoordinateGeneratorNormalizedLawDefOfScalarPushforward",
        "hBrownianCoordinateGeneratorNormalizedLawDef"
      ]
      status := ProofStatus.obligation
    },
    {
      id := "ASTIS.SALD.cycle184.lower_2_packet.normalized_law_scalar_pushforward"
      interface := "Compiled lower_2 bridge: derive hBrownianCoordinateGeneratorNormalizedLawDef from the scalar Brownian coordinate measurability hScalarMeas, the normalized law pushforward definition hNormalizedCoordinateLawDef, source integrand measurability hSourceTaylorIntegrandMeas, and the sample-space generator definition hGeneratorPullbackDef by MeasureTheory.integral_map."
      source := saldGeneralMovingTargetDiscreteEmSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.selectedWeakTestBrownianCoordinateGeneratorNormalizedLawDefOfScalarPushforward",
        "MeasureTheory.integral_map",
        "hScalarMeas",
        "hNormalizedCoordinateLawDef",
        "hSourceTaylorIntegrandMeas",
        "hGeneratorPullbackDef",
        "scalarBrownianCoordinate",
        "sourceTaylorIntegrand",
        "normalizedCoordinateLaw",
        "appendix.tex:984-995",
        "appendix.tex:1170-1176"
      ]
      reusedBy := [
        "hBrownianCoordinateGeneratorNormalizedLawDef",
        "SALD.selectedWeakTestBrownianCoordinateGeneratorSourceIntegralDefOfStdGaussianVectorLaw",
        "SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralAndAE",
        "sald.general_moving_target_discrete.em_interpolation_fp"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle184.remaining_brownian_taylor_backend"
      interface := "Remaining exact Brownian/Ito frozen backend after the lower_2 scalar-pushforward bridge: hScalarMeas, hNormalizedCoordinateLawDef, hSourceTaylorIntegrandMeas, hGeneratorPullbackDef, hSourceTaylorIntegrandPointwise, hRemainderGeneratorLimitDef, and hRemainderMeas/hRemainderBound/hRemainderBoundInt. hSourceHasHessian and hSourceHessianBound stay source-contract gaps."
      source := saldGeneralMovingTargetDiscreteEmSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "hScalarMeas",
        "hNormalizedCoordinateLawDef",
        "hSourceTaylorIntegrandMeas",
        "hGeneratorPullbackDef",
        "hSourceTaylorIntegrandPointwise",
        "hRemainderGeneratorLimitDef",
        "hRemainderMeas",
        "hRemainderBound",
        "hRemainderBoundInt",
        "hSourceHasHessian",
        "hSourceHessianBound",
        "source-contract-gap for selected weak-test Hessian fields"
      ]
      reusedBy := [
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.general_moving_target_discrete.kl_derivative",
        "divergence/FI/IBP handoff"
      ]
      status := ProofStatus.obligation
    }
  ]

/-! ### Cycle 185 lower_2: remainder generator normalized-law split -/

/-- Cycle-185 lower_2 packet for the remainder-generator law leaf.

The compiled theorem narrows `hRemainderGeneratorLimitDef` to the source
definition of the normalized remainder integral under the actual normalized
scalar-coordinate law, then reuses the already compiled Gaussian coordinate-law
and variance packaging bridges.  It is not a DCT/vanishing theorem and does not
touch the selected weak-test Hessian source-contract gap.
-/

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