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

AutoSamplingTheory.SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDag

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 cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDag :
    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.cycle161.middle_packet.coordinate_generator_from_one_dim_taylor
    interface : String (explicit)

    Text describing intended mathematical interface.

    Compiled theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDefOfOneDimTaylor derives hFrozenScalarBrownianItoCoordinateGeneratorDef from the smaller source-cited one-dimensional Brownian/Ito Taylor boundary hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor by unfolding SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator along the standard Brownian 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.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorLowerObligation
    2. SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator
    3. SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDefOfOneDimTaylor
    4. hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor
    5. hFrozenScalarBrownianItoCoordinateGeneratorDef
    6. SALD.gaussianRealZeroSecondMoment
    7. eq:general_moving_target_SALD_frozen_interp
    8. appendix.tex:984-995
    9. appendix.tex:1379-1387
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseDefOfCoordinateGenerator
    2. sald.general_moving_target_discrete.em_interpolation_fp
    3. sald.discrete_forward_kl.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.cycle161.lower_2_packet.one_dim_taylor_moment_collapse
    interface : String (explicit)

    Text describing intended mathematical interface.

    Compiled lower_2 theorem: SALD.gaussianRealZeroOneDimTaylorMomentContribution collapses the scalar Taylor linear/quadratic Gaussian moment contribution using ProbabilityTheory.integral_id_gaussianReal and SALD.gaussianRealZeroSecondMoment. The remaining source-cited gap below hFrozenScalarBrownianItoOneDimTaylorExpansion is now the dominated Taylor remainder and generator-limit theorem hFrozenScalarBrownianItoTaylorRemainderGeneratorLimit, not the centered moment algebra.
    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.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorMomentLowerObligation
    2. SALD.gaussianRealZeroOneDimTaylorMomentContribution
    3. SALD.gaussianRealZeroSecondMoment
    4. ProbabilityTheory.integral_id_gaussianReal
    5. hFrozenScalarBrownianItoOneDimTaylorExpansion
    6. hFrozenScalarBrownianItoTaylorRemainderGeneratorLimit
    7. eq:general_moving_target_SALD_frozen_interp
    8. appendix.tex:984-995
    9. appendix.tex:1379-1387
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor
    2. SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDefOfOneDimTaylor
    status : AutoSamplingTheory.ProofStatus (explicit)

    Recorded workflow status, default planned.

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

    Stable plan-node identifier.

    ASTIS.SALD.cycle161.remaining_one_dim_brownian_ito_taylor_boundary
    interface : String (explicit)

    Text describing intended mathematical interface.

    Remaining exact source boundary after the cycle-161 lower_2 moment split: prove hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor, identifying the scalar Brownian coordinate contribution with SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator for the selected weak test and standard Brownian basis direction. Cycle 161 lower_1 compiles SALD.gaussianRealZeroSecondMoment, and lower_2 compiles SALD.gaussianRealZeroOneDimTaylorMomentContribution, so the remaining smaller source-cited gap is the dominated Taylor remainder and generator-limit passage hFrozenScalarBrownianItoTaylorRemainderGeneratorLimit under selected-test C^2/integrability hypotheses and drift separation; sigma_eta^2/2 remains outside this event field.
    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.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorScoutObligation
    2. SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorMomentLowerObligation
    3. hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor
    4. hFrozenScalarBrownianItoOneDimTaylorExpansion
    5. hFrozenScalarBrownianItoTaylorRemainderGeneratorLimit
    6. SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator
    7. SALD.gaussianRealZeroSecondMoment
    8. SALD.gaussianRealZeroOneDimTaylorMomentContribution
    9. Mathlib.Probability.Distributions.Gaussian.Real
    10. Mathlib.Probability.Moments.CovarianceBilin
    11. Mathlib.Analysis.InnerProductSpace.Laplacian
    12. eq:general_moving_target_SALD_frozen_interp
    13. appendix.tex:984-995
    14. appendix.tex:1379-1387
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDefOfOneDimTaylor
    2. SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseDefOfCoordinateGenerator
    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.cycle161.residual_coordinate_sum_boundary
    interface : String (explicit)

    Text describing intended mathematical interface.

    Residual sibling source boundary: hFrozenScalarBrownianItoEventFieldCoordinateSum remains explicit and should only be discharged by a source proof that the abstract EM Laplacian event field is the sum of scalar Brownian coordinate generator contributions. This cycle does not hide it behind a downstream weak-FP or total-event consumer.
    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. hFrozenScalarBrownianItoEventFieldCoordinateSum
    2. eq:general_moving_target_SALD_frozen_interp
    3. appendix.tex:984-995
    4. appendix.tex:1379-1387
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseDefOfCoordinateGenerator
    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.cycle161.reviewer_one_dim_taylor_check
    interface : String (explicit)

    Text describing intended mathematical interface.

    Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hFrozenScalarBrownianItoCoordinateGeneratorDef is narrowed to hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor through the compiled cycle-161 theorem, hFrozenScalarBrownianItoEventFieldCoordinateSum remains explicit, and python3 tools/astis.py check passes without weak-FP source-field, trace-field, downstream total-event/source-functional, non-EM fallback, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, SLT import, or sald_version_2 use.
    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.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorLowerObligation
    2. SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorScoutObligation
    3. SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorMomentLowerObligation
    4. SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDefOfOneDimTaylor
    5. SALD.gaussianRealZeroSecondMoment
    6. SALD.gaussianRealZeroOneDimTaylorMomentContribution
    7. hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor
    8. hFrozenScalarBrownianItoEventFieldCoordinateSum
    9. appendix.tex:984-995
    10. appendix.tex:1379-1387
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. cycle 161 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 cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDag :
    List ProofDagBlock :=
  [
    {
      id := "ASTIS.SALD.cycle161.middle_packet.coordinate_generator_from_one_dim_taylor"
      interface := "Compiled theorem: SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDefOfOneDimTaylor derives hFrozenScalarBrownianItoCoordinateGeneratorDef from the smaller source-cited one-dimensional Brownian/Ito Taylor boundary hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor by unfolding SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator along the standard Brownian coordinate."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorLowerObligation",
        "SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator",
        "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDefOfOneDimTaylor",
        "hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor",
        "hFrozenScalarBrownianItoCoordinateGeneratorDef",
        "SALD.gaussianRealZeroSecondMoment",
        "eq:general_moving_target_SALD_frozen_interp",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseDefOfCoordinateGenerator",
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle161.lower_2_packet.one_dim_taylor_moment_collapse"
      interface := "Compiled lower_2 theorem: SALD.gaussianRealZeroOneDimTaylorMomentContribution collapses the scalar Taylor linear/quadratic Gaussian moment contribution using ProbabilityTheory.integral_id_gaussianReal and SALD.gaussianRealZeroSecondMoment. The remaining source-cited gap below hFrozenScalarBrownianItoOneDimTaylorExpansion is now the dominated Taylor remainder and generator-limit theorem hFrozenScalarBrownianItoTaylorRemainderGeneratorLimit, not the centered moment algebra."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorMomentLowerObligation",
        "SALD.gaussianRealZeroOneDimTaylorMomentContribution",
        "SALD.gaussianRealZeroSecondMoment",
        "ProbabilityTheory.integral_id_gaussianReal",
        "hFrozenScalarBrownianItoOneDimTaylorExpansion",
        "hFrozenScalarBrownianItoTaylorRemainderGeneratorLimit",
        "eq:general_moving_target_SALD_frozen_interp",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor",
        "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDefOfOneDimTaylor"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle161.remaining_one_dim_brownian_ito_taylor_boundary"
      interface := "Remaining exact source boundary after the cycle-161 lower_2 moment split: prove hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor, identifying the scalar Brownian coordinate contribution with SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator for the selected weak test and standard Brownian basis direction. Cycle 161 lower_1 compiles SALD.gaussianRealZeroSecondMoment, and lower_2 compiles SALD.gaussianRealZeroOneDimTaylorMomentContribution, so the remaining smaller source-cited gap is the dominated Taylor remainder and generator-limit passage hFrozenScalarBrownianItoTaylorRemainderGeneratorLimit under selected-test C^2/integrability hypotheses and drift separation; sigma_eta^2/2 remains outside this event field."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorScoutObligation",
        "SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorMomentLowerObligation",
        "hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor",
        "hFrozenScalarBrownianItoOneDimTaylorExpansion",
        "hFrozenScalarBrownianItoTaylorRemainderGeneratorLimit",
        "SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator",
        "SALD.gaussianRealZeroSecondMoment",
        "SALD.gaussianRealZeroOneDimTaylorMomentContribution",
        "Mathlib.Probability.Distributions.Gaussian.Real",
        "Mathlib.Probability.Moments.CovarianceBilin",
        "Mathlib.Analysis.InnerProductSpace.Laplacian",
        "eq:general_moving_target_SALD_frozen_interp",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDefOfOneDimTaylor",
        "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseDefOfCoordinateGenerator"
      ]
      status := ProofStatus.obligation
    },
    {
      id := "ASTIS.SALD.cycle161.residual_coordinate_sum_boundary"
      interface := "Residual sibling source boundary: hFrozenScalarBrownianItoEventFieldCoordinateSum remains explicit and should only be discharged by a source proof that the abstract EM Laplacian event field is the sum of scalar Brownian coordinate generator contributions. This cycle does not hide it behind a downstream weak-FP or total-event consumer."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "hFrozenScalarBrownianItoEventFieldCoordinateSum",
        "eq:general_moving_target_SALD_frozen_interp",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseDefOfCoordinateGenerator"
      ]
      status := ProofStatus.obligation
    },
    {
      id := "ASTIS.SALD.cycle161.reviewer_one_dim_taylor_check"
      interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hFrozenScalarBrownianItoCoordinateGeneratorDef is narrowed to hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor through the compiled cycle-161 theorem, hFrozenScalarBrownianItoEventFieldCoordinateSum remains explicit, and python3 tools/astis.py check passes without weak-FP source-field, trace-field, downstream total-event/source-functional, non-EM fallback, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, SLT import, or sald_version_2 use."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorLowerObligation",
        "SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorScoutObligation",
        "SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorMomentLowerObligation",
        "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDefOfOneDimTaylor",
        "SALD.gaussianRealZeroSecondMoment",
        "SALD.gaussianRealZeroOneDimTaylorMomentContribution",
        "hFrozenScalarBrownianItoCoordinateGeneratorOneDimTaylor",
        "hFrozenScalarBrownianItoEventFieldCoordinateSum",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := ["cycle 161 reviewer"]
      status := ProofStatus.obligation
    }
  ]

/-! ### Cycle 162: scalar Brownian Ito Taylor remainder split -/

/-- Cycle-162 narrowing below the one-dimensional Brownian/Ito Taylor boundary.

The remaining cycle-161 source theorem
`hFrozenScalarBrownianItoTaylorRemainderGeneratorLimit` is no longer treated as
one opaque generator equality.  The compiled bridge below consumes the Gaussian
moment collapse and leaves only the source Taylor moment decomposition,
quadratic-variation normalization, and normalized Taylor-remainder vanishing
as explicit analytic obligations.
-/

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