AutoSamplingTheory.SALD.cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderQuadraticBoundDag

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 cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderQuadraticBoundDag :
    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.cycle165.lower_packet.normalized_remainder_hbound_from_taylor_quotient
interface : String (explicit)
Text describing intended mathematical interface.
Compiled theorem: SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound supplies the DCT hBound input for the concrete selected-test normalized Taylor remainder from a deterministic scalar Taylor quotient bound along r |-> sourceTest (x + r • e).
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.cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderQuadraticBoundLowerObligation
2. SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound
3. SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainder
4. hTaylorQuotientBound
5. hBound
6. eq:general_moving_target_SALD_frozen_interp
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.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero
2. hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes
3. 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.cycle165.lower_2_packet.taylor_quotient_split_first_order_second_coeff
interface : String (explicit)
Text describing intended mathematical interface.
Compiled theorem: SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff supplies the deterministic selected-line second-order Taylor quotient bound from a first-order quadratic remainder estimate, a second Taylor coefficient bound, and constant bookkeeping.
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.cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorQuotientSplitLower2Obligation
2. SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff
3. hFirst
4. hSecondCoeff
5. taylorWithinEval_succ
6. taylorCoeffWithin
7. norm_sub_le
8. eq:general_moving_target_SALD_frozen_interp
9. appendix.tex:984-995
10. appendix.tex:1379-1387
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. hTaylorQuotientBound
2. SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound
3. SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero
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.cycle165.remaining_taylor_quotient_bound
interface : String (explicit)
Text describing intended mathematical interface.
Remaining exact theorem after the lower_2 split: prove the smaller source-cited first-order quadratic Taylor remainder estimate hFirst and the selected-line second Taylor coefficient bound hSecondCoeff, with constant bookkeeping C1 + C2 <= C. Keep the Taylor moment decomposition, Brownian quadratic-variation normalization, and coordinate-sum leaves 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. SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff
2. hFirst
3. hSecondCoeff
4. selected-test regularity or compact-support/bounded-Hessian source hypothesis
5. SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound
6. SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderEventuallyAEStronglyMeasurable
7. SALD.gaussianRealSelectedTestLineSecondOrderQuadraticBoundIntegrable
8. hFrozenScalarBrownianItoTaylorMomentDecomposition
9. hFrozenScalarBrownianItoQuadraticVariationNormalization
10. hFrozenScalarBrownianItoEventFieldCoordinateSum
11. eq:general_moving_target_SALD_frozen_interp
12. appendix.tex:984-995
13. appendix.tex:1379-1387
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero
2. hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes
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.cycle165.reviewer_hbound_check
interface : String (explicit)
Text describing intended mathematical interface.
Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, the compiled theorem targets the concrete hBound domination shape rather than a new wrapper, the remaining source theorem is the deterministic Taylor quotient bound, 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.cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderQuadraticBoundLowerObligation
2. SALD.cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorQuotientSplitLower2Obligation
3. SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound
4. SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff
5. appendix.tex:984-995
6. appendix.tex:1379-1387
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. cycle 165 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 cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderQuadraticBoundDag :
    List ProofDagBlock :=
  [
    {
      id := "ASTIS.SALD.cycle165.lower_packet.normalized_remainder_hbound_from_taylor_quotient"
      interface := "Compiled theorem: SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound supplies the DCT hBound input for the concrete selected-test normalized Taylor remainder from a deterministic scalar Taylor quotient bound along r |-> sourceTest (x + r • e)."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderQuadraticBoundLowerObligation",
        "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound",
        "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainder",
        "hTaylorQuotientBound",
        "hBound",
        "eq:general_moving_target_SALD_frozen_interp",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero",
        "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes",
        "SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle165.lower_2_packet.taylor_quotient_split_first_order_second_coeff"
      interface := "Compiled theorem: SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff supplies the deterministic selected-line second-order Taylor quotient bound from a first-order quadratic remainder estimate, a second Taylor coefficient bound, and constant bookkeeping."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorQuotientSplitLower2Obligation",
        "SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff",
        "hFirst",
        "hSecondCoeff",
        "taylorWithinEval_succ",
        "taylorCoeffWithin",
        "norm_sub_le",
        "eq:general_moving_target_SALD_frozen_interp",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "hTaylorQuotientBound",
        "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound",
        "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle165.remaining_taylor_quotient_bound"
      interface := "Remaining exact theorem after the lower_2 split: prove the smaller source-cited first-order quadratic Taylor remainder estimate hFirst and the selected-line second Taylor coefficient bound hSecondCoeff, with constant bookkeeping C1 + C2 <= C. Keep the Taylor moment decomposition, Brownian quadratic-variation normalization, and coordinate-sum leaves separate."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff",
        "hFirst",
        "hSecondCoeff",
        "selected-test regularity or compact-support/bounded-Hessian source hypothesis",
        "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound",
        "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderEventuallyAEStronglyMeasurable",
        "SALD.gaussianRealSelectedTestLineSecondOrderQuadraticBoundIntegrable",
        "hFrozenScalarBrownianItoTaylorMomentDecomposition",
        "hFrozenScalarBrownianItoQuadraticVariationNormalization",
        "hFrozenScalarBrownianItoEventFieldCoordinateSum",
        "eq:general_moving_target_SALD_frozen_interp",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero",
        "hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes"
      ]
      status := ProofStatus.obligation
    },
    {
      id := "ASTIS.SALD.cycle165.reviewer_hbound_check"
      interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, the compiled theorem targets the concrete hBound domination shape rather than a new wrapper, the remaining source theorem is the deterministic Taylor quotient bound, and python3 tools/astis.py check passes."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderQuadraticBoundLowerObligation",
        "SALD.cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorQuotientSplitLower2Obligation",
        "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound",
        "SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := ["cycle 165 reviewer"]
      status := ProofStatus.obligation
    }
  ]

/-! ### Cycle 166: first-order selected-line quadratic remainder quotient -/

/-- Cycle-166 dynamic-leaf packet for the selected-line first-order remainder.

The cycle-165 split left `hFirst` as a supplied deterministic quotient bound.
This packet narrows that supplied quotient to the source-facing non-quotient
quadratic Taylor-remainder estimate for the degree-one selected scalar line.
-/

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