Samplinglib
Lean gate not recorded for this source state main · 0e31a3cda412

AutoSamplingTheory.SALD.cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedVarianceDag

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 cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedVarianceDag :
    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

  1. id : String (explicit)

    Stable plan-node identifier.

    ASTIS.SALD.cycle178.middle_packet.normalized_variance_from_gaussian_law
    interface : String (explicit)

    Text describing intended mathematical interface.

    Compiled bridge: SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw derives hNormalizedVarianceDef from a source-backed normalized scalar coordinate law hNormalizedCoordinateLaw and the local variance-field definition hVarianceDef, using Mathlib's Gaussian variance theorem.
    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.cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedVarianceMiddleObligation
    2. SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw
    3. ProbabilityTheory.variance_id_gaussianReal
    4. NNReal.coe_injective
    5. hNormalizedVarianceDef
    6. hNormalizedCoordinateLaw
    7. hVarianceDef
    8. appendix.tex:958-970
    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. SALD.selectedWeakTestVarianceOneOfNormalizedBrownianVarianceDef
    2. SALD.selectedWeakTestQuadraticVariationNormalizationOfScalarLineSecondCoeffAndNormalizedVarianceDef
    3. hFrozenScalarBrownianItoQuadraticVariationNormalization
    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
  2. id : String (explicit)

    Stable plan-node identifier.

    ASTIS.SALD.cycle178.lower_1_packet.normalized_coordinate_law_projection_route
    interface : String (explicit)

    Text describing intended mathematical interface.

    Lower_1 proof-scout route: narrow hNormalizedCoordinateLaw to the paper's normalized vector standard-Gaussian law hNormalizedVectorLaw plus the coordinate-law definition hCoordinateLawDef. Lower_2 should prove the standard-Gaussian coordinate projection with Mathlib's stdGaussian/IsGaussian API, then rewrite the two source-backed fields.
    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.cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedCoordinateLawLower1Obligation
    2. hNormalizedVectorLaw
    3. hCoordinateLawDef
    4. hNormalizedCoordinateLaw
    5. ProbabilityTheory.stdGaussian
    6. ProbabilityTheory.IsGaussian.map_eq_gaussianReal
    7. ProbabilityTheory.integral_strongDual_stdGaussian
    8. ProbabilityTheory.variance_dual_stdGaussian
    9. InnerProductSpace.toDual
    10. stdOrthonormalBasis
    11. appendix.tex:958-970
    12. appendix.tex:984-995
    13. appendix.tex:1170-1176
    14. appendix.tex:1379-1387
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. hNormalizedCoordinateLaw
    2. hNormalizedVarianceDef
    3. SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw
    4. Brownian/Ito scalar generator bookkeeping
    status : AutoSamplingTheory.ProofStatus (explicit)

    Recorded workflow status, default planned.

    AutoSamplingTheory.ProofStatus.obligation — stored label only; no proof certification
  3. id : String (explicit)

    Stable plan-node identifier.

    ASTIS.SALD.cycle178.lower_2_packet.normalized_coordinate_law_from_vector_gaussian
    interface : String (explicit)

    Text describing intended mathematical interface.

    Compiled lower_2 bridge: SALD.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw derives hNormalizedCoordinateLaw from the source-backed vector standard-Gaussian normalized increment law hNormalizedVectorLaw plus hCoordinateLawDef by projecting stdGaussian along the standard orthonormal coordinate.
    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.cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedCoordinateLawLower2Obligation
    2. SALD.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw
    3. hNormalizedVectorLaw
    4. hCoordinateLawDef
    5. hNormalizedCoordinateLaw
    6. ProbabilityTheory.stdGaussian
    7. ProbabilityTheory.IsGaussian.map_eq_gaussianReal
    8. ProbabilityTheory.integral_strongDual_stdGaussian
    9. ProbabilityTheory.variance_dual_stdGaussian
    10. InnerProductSpace.toDual
    11. stdOrthonormalBasis
    12. appendix.tex:958-970
    13. appendix.tex:984-995
    14. appendix.tex:1170-1176
    15. appendix.tex:1379-1387
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw
    2. hNormalizedVarianceDef
    3. hFrozenScalarBrownianItoQuadraticVariationNormalization
    4. Brownian/Ito scalar generator bookkeeping
    status : AutoSamplingTheory.ProofStatus (explicit)

    Recorded workflow status, default planned.

    AutoSamplingTheory.ProofStatus.formalized — stored label only; no proof certification
  4. id : String (explicit)

    Stable plan-node identifier.

    ASTIS.SALD.cycle178.remaining_normalized_coordinate_law_boundary
    interface : String (explicit)

    Text describing intended mathematical interface.

    Remaining exact variance-side source boundary: prove the vector standard-Gaussian normalized increment law hNormalizedVectorLaw, the coordinate-law definition hCoordinateLawDef, and the local NNReal variance-field packaging hVarianceDef. Keep hScalarLineSecondCoeffDef and source-Hessian fields 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. hNormalizedVectorLaw
    2. hCoordinateLawDef
    3. hVarianceDef
    4. hScalarLineSecondCoeffDef
    5. hSourceHasHessian
    6. hSourceHessianBound
    7. hSecondCoeff
    8. hFrozenScalarBrownianItoTaylorMomentDecomposition
    9. hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes
    10. hFrozenScalarBrownianItoEventFieldCoordinateSum
    11. appendix.tex:958-970
    12. appendix.tex:984-995
    13. appendix.tex:1170-1176
    14. appendix.tex:1379-1387
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. hNormalizedVarianceDef
    2. hFrozenScalarBrownianItoQuadraticVariationNormalization
    3. Brownian/Ito scalar generator bookkeeping
    status : AutoSamplingTheory.ProofStatus (explicit)

    Recorded workflow status, default planned.

    AutoSamplingTheory.ProofStatus.obligation — stored label only; no proof certification
  5. id : String (explicit)

    Stable plan-node identifier.

    ASTIS.SALD.cycle178.reviewer_normalized_variance_check
    interface : String (explicit)

    Text describing intended mathematical interface.

    Reviewer accepts only if the packet is narrows-source-cited-boundary, hNormalizedVarianceDef is narrowed to a source-backed Gaussian coordinate law rather than restated, sigma_eta^2/2 remains outside the event field, source-Hessian fields remain source gaps, sibling Brownian/Ito leaves remain explicit, and python3 tools/astis.py check passes.
    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.cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedVarianceMiddleObligation
    2. SALD.cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedCoordinateLawLower2Obligation
    3. SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw
    4. SALD.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw
    5. hNormalizedVarianceDef
    6. hNormalizedCoordinateLaw
    7. hNormalizedVectorLaw
    8. hCoordinateLawDef
    9. hVarianceDef
    10. hSourceHasHessian
    11. hSourceHessianBound
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. cycle 178 reviewer
    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 cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedVarianceDag :
    List ProofDagBlock :=
  [
    {
      id := "ASTIS.SALD.cycle178.middle_packet.normalized_variance_from_gaussian_law"
      interface := "Compiled bridge: SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw derives hNormalizedVarianceDef from a source-backed normalized scalar coordinate law hNormalizedCoordinateLaw and the local variance-field definition hVarianceDef, using Mathlib's Gaussian variance theorem."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedVarianceMiddleObligation",
        "SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw",
        "ProbabilityTheory.variance_id_gaussianReal",
        "NNReal.coe_injective",
        "hNormalizedVarianceDef",
        "hNormalizedCoordinateLaw",
        "hVarianceDef",
        "appendix.tex:958-970",
        "appendix.tex:984-995",
        "appendix.tex:1170-1176",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "SALD.selectedWeakTestVarianceOneOfNormalizedBrownianVarianceDef",
        "SALD.selectedWeakTestQuadraticVariationNormalizationOfScalarLineSecondCoeffAndNormalizedVarianceDef",
        "hFrozenScalarBrownianItoQuadraticVariationNormalization",
        "sald.general_moving_target_discrete.em_interpolation_fp"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle178.lower_1_packet.normalized_coordinate_law_projection_route"
      interface := "Lower_1 proof-scout route: narrow hNormalizedCoordinateLaw to the paper's normalized vector standard-Gaussian law hNormalizedVectorLaw plus the coordinate-law definition hCoordinateLawDef. Lower_2 should prove the standard-Gaussian coordinate projection with Mathlib's stdGaussian/IsGaussian API, then rewrite the two source-backed fields."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedCoordinateLawLower1Obligation",
        "hNormalizedVectorLaw",
        "hCoordinateLawDef",
        "hNormalizedCoordinateLaw",
        "ProbabilityTheory.stdGaussian",
        "ProbabilityTheory.IsGaussian.map_eq_gaussianReal",
        "ProbabilityTheory.integral_strongDual_stdGaussian",
        "ProbabilityTheory.variance_dual_stdGaussian",
        "InnerProductSpace.toDual",
        "stdOrthonormalBasis",
        "appendix.tex:958-970",
        "appendix.tex:984-995",
        "appendix.tex:1170-1176",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "hNormalizedCoordinateLaw",
        "hNormalizedVarianceDef",
        "SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw",
        "Brownian/Ito scalar generator bookkeeping"
      ]
      status := ProofStatus.obligation
    },
    {
      id := "ASTIS.SALD.cycle178.lower_2_packet.normalized_coordinate_law_from_vector_gaussian"
      interface := "Compiled lower_2 bridge: SALD.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw derives hNormalizedCoordinateLaw from the source-backed vector standard-Gaussian normalized increment law hNormalizedVectorLaw plus hCoordinateLawDef by projecting stdGaussian along the standard orthonormal coordinate."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedCoordinateLawLower2Obligation",
        "SALD.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw",
        "hNormalizedVectorLaw",
        "hCoordinateLawDef",
        "hNormalizedCoordinateLaw",
        "ProbabilityTheory.stdGaussian",
        "ProbabilityTheory.IsGaussian.map_eq_gaussianReal",
        "ProbabilityTheory.integral_strongDual_stdGaussian",
        "ProbabilityTheory.variance_dual_stdGaussian",
        "InnerProductSpace.toDual",
        "stdOrthonormalBasis",
        "appendix.tex:958-970",
        "appendix.tex:984-995",
        "appendix.tex:1170-1176",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw",
        "hNormalizedVarianceDef",
        "hFrozenScalarBrownianItoQuadraticVariationNormalization",
        "Brownian/Ito scalar generator bookkeeping"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle178.remaining_normalized_coordinate_law_boundary"
      interface := "Remaining exact variance-side source boundary: prove the vector standard-Gaussian normalized increment law hNormalizedVectorLaw, the coordinate-law definition hCoordinateLawDef, and the local NNReal variance-field packaging hVarianceDef. Keep hScalarLineSecondCoeffDef and source-Hessian fields separate."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "hNormalizedVectorLaw",
        "hCoordinateLawDef",
        "hVarianceDef",
        "hScalarLineSecondCoeffDef",
        "hSourceHasHessian",
        "hSourceHessianBound",
        "hSecondCoeff",
        "hFrozenScalarBrownianItoTaylorMomentDecomposition",
        "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes",
        "hFrozenScalarBrownianItoEventFieldCoordinateSum",
        "appendix.tex:958-970",
        "appendix.tex:984-995",
        "appendix.tex:1170-1176",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "hNormalizedVarianceDef",
        "hFrozenScalarBrownianItoQuadraticVariationNormalization",
        "Brownian/Ito scalar generator bookkeeping"
      ]
      status := ProofStatus.obligation
    },
    {
      id := "ASTIS.SALD.cycle178.reviewer_normalized_variance_check"
      interface := "Reviewer accepts only if the packet is narrows-source-cited-boundary, hNormalizedVarianceDef is narrowed to a source-backed Gaussian coordinate law rather than restated, sigma_eta^2/2 remains outside the event field, source-Hessian fields remain source gaps, sibling Brownian/Ito leaves remain explicit, and python3 tools/astis.py check passes."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedVarianceMiddleObligation",
        "SALD.cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedCoordinateLawLower2Obligation",
        "SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw",
        "SALD.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw",
        "hNormalizedVarianceDef",
        "hNormalizedCoordinateLaw",
        "hNormalizedVectorLaw",
        "hCoordinateLawDef",
        "hVarianceDef",
        "hSourceHasHessian",
        "hSourceHessianBound"
      ]
      reusedBy := ["cycle 178 reviewer"]
      status := ProofStatus.obligation
    }
  ]

/-! ### Cycle 179 middle: Hessian audit closure and scalar-line coefficient packet -/

/-- Cycle-179 middle packet after the source-Hessian audit.

The selected weak-test Hessian fields are not derivable from the checked
original-source anchors.  This packet keeps those fields as source-contract
gaps and assigns the next connected Brownian/Ito leaf, the scalar-line second
Taylor coefficient boundary.
-/

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