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

AutoSamplingTheory.SALD.cycle175GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorDominationDag

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 cycle175GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorDominationDag :
    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.cycle175.middle_packet.std_basis_first_order_remainder_from_source_hessian
    interface : String (explicit)

    Text describing intended mathematical interface.

    Compiled theorem: SALD.gaussianRealSelectedTestStdOrthonormalFirstOrderQuadraticRemainderBoundOfSourceHessianField supplies the first-order quadratic Taylor remainder for the selected scalar line in a standard Brownian coordinate from ambient ContDiffOn plus the source-backed Hessian representative 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.cycle175GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorDominationMiddleObligation
    2. SALD.gaussianRealSelectedTestStdOrthonormalFirstOrderQuadraticRemainderBoundOfSourceHessianField
    3. SALD.selectedWeakTestHessianOpNormOfSourceHessianField
    4. SALD.gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm
    5. SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNormStdOrthonormalBasis
    6. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound
    7. hSource
    8. hSourceHasHessian
    9. hSourceHessianBound
    10. appendix.tex:984-995
    11. appendix.tex:1026-1072
    12. appendix.tex:1379-1387
    13. main_body.tex:273-305
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. hFirst
    2. SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff
    3. SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound
    4. hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes
    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.cycle175.remaining_selected_line_taylor_boundary
    interface : String (explicit)

    Text describing intended mathematical interface.

    Remaining exact boundary after the cycle-175 bridge: source the selected weak-test ambient ContDiffOn field and the two source-Hessian fields hSourceHasHessian/hSourceHessianBound, then separately source hSecondCoeff before assembling the second-order quotient bound. Do not replace these fields by testRegular wrappers or 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. hSource
    2. hSourceHasHessian
    3. hSourceHessianBound
    4. hSecondCoeff
    5. hQuadraticCoeffDef
    6. hVarianceOne
    7. appendix.tex:984-995
    8. appendix.tex:1026-1072
    9. appendix.tex:1379-1387
    10. main_body.tex:273-305
    11. source-contract-gap for selected weak-test Hessian fields
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. normalized-remainder hBound
    2. Brownian/Ito scalar generator bookkeeping
    3. sald.general_moving_target_discrete.em_interpolation_fp
    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.cycle175.lower_1_packet.quadratic_coeff_second_taylor_route
    interface : String (explicit)

    Text describing intended mathematical interface.

    Lower_1 proof-scout route: narrow hQuadraticCoeffDef to hSecondTaylorCoeffDef, the source-backed identity that the normalized scalar Brownian quadratic coefficient equals the diagonal second Frechet derivative in the standard Brownian coordinate. Lower_2 can then unfold SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator to recover hQuadraticCoeffDef.
    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.cycle175GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoQuadraticCoeffLower1Obligation
    2. SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator
    3. hQuadraticCoeffDef
    4. hSecondTaylorCoeffDef
    5. hVarianceOne
    6. appendix.tex:984-995
    7. appendix.tex:1379-1387
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. SALD.selectedWeakTestQuadraticVariationNormalizationOfCoeffDefAndVarianceOne
    2. hFrozenScalarBrownianItoQuadraticVariationNormalization
    3. sald.general_moving_target_discrete.em_interpolation_fp
    status : AutoSamplingTheory.ProofStatus (explicit)

    Recorded workflow status, default planned.

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

    Stable plan-node identifier.

    ASTIS.SALD.cycle175.lower_2_packet.quadratic_coeff_second_taylor_bridge
    interface : String (explicit)

    Text describing intended mathematical interface.

    Compiled lower_2 bridge: SALD.selectedWeakTestQuadraticCoeffDefOfSecondTaylorCoeffDef derives hQuadraticCoeffDef by unfolding SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator from the source-backed hSecondTaylorCoeffDef identity.
    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.selectedWeakTestQuadraticCoeffDefOfSecondTaylorCoeffDef
    2. SALD.cycle175GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoQuadraticCoeffLower2Obligation
    3. SALD.cycle175GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoQuadraticCoeffLower1Obligation
    4. SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator
    5. hSecondTaylorCoeffDef
    6. hQuadraticCoeffDef
    7. appendix.tex:984-995
    8. appendix.tex:1379-1387
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. SALD.selectedWeakTestQuadraticVariationNormalizationOfCoeffDefAndVarianceOne
    2. hFrozenScalarBrownianItoQuadraticVariationNormalization
    3. 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
  5. id : String (explicit)

    Stable plan-node identifier.

    ASTIS.SALD.cycle175.reviewer_taylor_domination_check
    interface : String (explicit)

    Text describing intended mathematical interface.

    Reviewer accepts only if the packet is narrows-source-cited-boundary, the compiled theorems narrow the selected-line Taylor-domination and quadratic-coefficient boundaries rather than adding broad wrappers, hSourceHasHessian/hSourceHessianBound and hSecondTaylorCoeffDef remain source-facing 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.cycle175GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorDominationMiddleObligation
    2. SALD.cycle175GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoQuadraticCoeffLower2Obligation
    3. ASTIS.SALD.cycle175.middle_packet.std_basis_first_order_remainder_from_source_hessian
    4. ASTIS.SALD.cycle175.remaining_selected_line_taylor_boundary
    5. ASTIS.SALD.cycle175.lower_2_packet.quadratic_coeff_second_taylor_bridge
    6. SALD.selectedWeakTestQuadraticCoeffDefOfSecondTaylorCoeffDef
    7. SALD.gaussianRealSelectedTestStdOrthonormalFirstOrderQuadraticRemainderBoundOfSourceHessianField
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. cycle 175 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 cycle175GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorDominationDag :
    List ProofDagBlock :=
  [
    {
      id := "ASTIS.SALD.cycle175.middle_packet.std_basis_first_order_remainder_from_source_hessian"
      interface := "Compiled theorem: SALD.gaussianRealSelectedTestStdOrthonormalFirstOrderQuadraticRemainderBoundOfSourceHessianField supplies the first-order quadratic Taylor remainder for the selected scalar line in a standard Brownian coordinate from ambient ContDiffOn plus the source-backed Hessian representative fields."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle175GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorDominationMiddleObligation",
        "SALD.gaussianRealSelectedTestStdOrthonormalFirstOrderQuadraticRemainderBoundOfSourceHessianField",
        "SALD.selectedWeakTestHessianOpNormOfSourceHessianField",
        "SALD.gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm",
        "SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNormStdOrthonormalBasis",
        "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound",
        "hSource",
        "hSourceHasHessian",
        "hSourceHessianBound",
        "appendix.tex:984-995",
        "appendix.tex:1026-1072",
        "appendix.tex:1379-1387",
        "main_body.tex:273-305"
      ]
      reusedBy := [
        "hFirst",
        "SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff",
        "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound",
        "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle175.remaining_selected_line_taylor_boundary"
      interface := "Remaining exact boundary after the cycle-175 bridge: source the selected weak-test ambient ContDiffOn field and the two source-Hessian fields hSourceHasHessian/hSourceHessianBound, then separately source hSecondCoeff before assembling the second-order quotient bound. Do not replace these fields by testRegular wrappers or VP score-Hessian regularity."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "hSource",
        "hSourceHasHessian",
        "hSourceHessianBound",
        "hSecondCoeff",
        "hQuadraticCoeffDef",
        "hVarianceOne",
        "appendix.tex:984-995",
        "appendix.tex:1026-1072",
        "appendix.tex:1379-1387",
        "main_body.tex:273-305",
        "source-contract-gap for selected weak-test Hessian fields"
      ]
      reusedBy := [
        "normalized-remainder hBound",
        "Brownian/Ito scalar generator bookkeeping",
        "sald.general_moving_target_discrete.em_interpolation_fp"
      ]
      status := ProofStatus.obligation
    },
    {
      id := "ASTIS.SALD.cycle175.lower_1_packet.quadratic_coeff_second_taylor_route"
      interface := "Lower_1 proof-scout route: narrow hQuadraticCoeffDef to hSecondTaylorCoeffDef, the source-backed identity that the normalized scalar Brownian quadratic coefficient equals the diagonal second Frechet derivative in the standard Brownian coordinate. Lower_2 can then unfold SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator to recover hQuadraticCoeffDef."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle175GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoQuadraticCoeffLower1Obligation",
        "SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator",
        "hQuadraticCoeffDef",
        "hSecondTaylorCoeffDef",
        "hVarianceOne",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "SALD.selectedWeakTestQuadraticVariationNormalizationOfCoeffDefAndVarianceOne",
        "hFrozenScalarBrownianItoQuadraticVariationNormalization",
        "sald.general_moving_target_discrete.em_interpolation_fp"
      ]
      status := ProofStatus.obligation
    },
    {
      id := "ASTIS.SALD.cycle175.lower_2_packet.quadratic_coeff_second_taylor_bridge"
      interface := "Compiled lower_2 bridge: SALD.selectedWeakTestQuadraticCoeffDefOfSecondTaylorCoeffDef derives hQuadraticCoeffDef by unfolding SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator from the source-backed hSecondTaylorCoeffDef identity."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.selectedWeakTestQuadraticCoeffDefOfSecondTaylorCoeffDef",
        "SALD.cycle175GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoQuadraticCoeffLower2Obligation",
        "SALD.cycle175GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoQuadraticCoeffLower1Obligation",
        "SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator",
        "hSecondTaylorCoeffDef",
        "hQuadraticCoeffDef",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "SALD.selectedWeakTestQuadraticVariationNormalizationOfCoeffDefAndVarianceOne",
        "hFrozenScalarBrownianItoQuadraticVariationNormalization",
        "sald.general_moving_target_discrete.em_interpolation_fp"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle175.reviewer_taylor_domination_check"
      interface := "Reviewer accepts only if the packet is narrows-source-cited-boundary, the compiled theorems narrow the selected-line Taylor-domination and quadratic-coefficient boundaries rather than adding broad wrappers, hSourceHasHessian/hSourceHessianBound and hSecondTaylorCoeffDef remain source-facing gaps, sibling Brownian/Ito leaves remain explicit, and python3 tools/astis.py check passes."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle175GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorDominationMiddleObligation",
        "SALD.cycle175GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoQuadraticCoeffLower2Obligation",
        "ASTIS.SALD.cycle175.middle_packet.std_basis_first_order_remainder_from_source_hessian",
        "ASTIS.SALD.cycle175.remaining_selected_line_taylor_boundary",
        "ASTIS.SALD.cycle175.lower_2_packet.quadratic_coeff_second_taylor_bridge",
        "SALD.selectedWeakTestQuadraticCoeffDefOfSecondTaylorCoeffDef",
        "SALD.gaussianRealSelectedTestStdOrthonormalFirstOrderQuadraticRemainderBoundOfSourceHessianField"
      ]
      reusedBy := ["cycle 175 reviewer"]
      status := ProofStatus.obligation
    }
  ]

/-! ### Cycle 176: normalized Brownian variance source field -/

/-- Cycle-176 dynamic-leaf worker packet for the normalized scalar Brownian
variance field.

The cycle-175 reviewer accepted the quadratic-coefficient bridge, leaving
`hSecondTaylorCoeffDef` plus the separate `hVarianceOne` field.  This packet
narrows only the variance side: the downstream real-valued `hVarianceOne`
hypothesis follows from the source-facing definition that the normalized scalar
Brownian coordinate has variance `1 : NNReal`.
-/

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