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

AutoSamplingTheory.SALD.cycle169GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDirectionalSecondDag

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 cycle169GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDirectionalSecondDag :
    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.cycle169.middle_packet.line_second_from_ambient_directional_hessian
    interface : String (explicit)

    Text describing intended mathematical interface.

    Compiled theorem: SALD.gaussianRealSelectedTestLineSecondDerivEqDirectionalFDeriv rewrites the global scalar-line iteratedDeriv 2 as the ambient selected-test second Frechet derivative applied twice to the affine-line direction 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.gaussianRealSelectedTestLineSecondDerivEqDirectionalFDeriv
    2. hSource
    3. iteratedDeriv_vcomp_two
    4. iteratedDeriv_const_add
    5. iteratedDeriv_smul_const
    6. iteratedDeriv_fun_id
    7. deriv_smul_const
    8. appendix.tex:984-995
    9. appendix.tex:1379-1387
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. hLineSecond
    2. hNegLineSecond
    3. SALD.gaussianRealSelectedTestLineSecondBoundsOfDirectionalSecondBound
    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.cycle169.middle_packet.reflected_line_second_from_same_bound
    interface : String (explicit)

    Text describing intended mathematical interface.

    Compiled theorem: SALD.gaussianRealSelectedTestLineSecondBoundsOfDirectionalSecondBound derives both hLineSecond and hNegLineSecond from one ambient diagonal directional-Hessian bound, using ContinuousMultilinearMap.map_smul_univ to cancel the two reflected signs.
    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.gaussianRealSelectedTestLineSecondBoundsOfDirectionalSecondBound
    2. SALD.gaussianRealSelectedTestLineSecondDerivEqDirectionalFDeriv
    3. hDirectionalSecond
    4. hLineSecond
    5. hNegLineSecond
    6. ContinuousMultilinearMap.map_smul_univ
    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.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound
    2. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndLineSecondBounds
    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.cycle169.lower_2_packet.directional_hessian_from_second_fderiv_op_norm
    interface : String (explicit)

    Text describing intended mathematical interface.

    Compiled theorem: SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNorm supplies hDirectionalSecond from a uniform selected-test second-Frechet-derivative operator-norm bound and the unit-direction side condition norm e <= 1.
    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.cycle169GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDirectionalSecondLower2Obligation
    2. SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNorm
    3. hSecondFDerivOpNorm
    4. heUnit
    5. ContinuousMultilinearMap.le_mul_prod_of_opNorm_le_of_le
    6. Finset.prod_const
    7. appendix.tex:984-995
    8. appendix.tex:1379-1387
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. hDirectionalSecond
    2. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound
    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.cycle169.middle_packet.directional_hessian_consumer
    interface : String (explicit)

    Text describing intended mathematical interface.

    Compiled theorem: SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound feeds the ambient directional-Hessian bound into the accepted cycle-168 Taylor bridge and derives hC1 from the same norm bound.
    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.cycle169GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDirectionalSecondMiddleObligation
    2. SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNorm
    3. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound
    4. SALD.gaussianRealSelectedTestLineSecondBoundsOfDirectionalSecondBound
    5. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndLineSecondBounds
    6. hDirectionalSecond
    7. hSource
    8. hC1
    9. appendix.tex:984-995
    10. appendix.tex:1379-1387
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. hFirst
    2. SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff
    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.cycle169.remaining_second_fderiv_operator_norm_bound
    interface : String (explicit)

    Text describing intended mathematical interface.

    Remaining exact theorem boundary after cycle 169 lower_2: expose the selected-test second-Frechet-derivative operator-norm bound forall z, norm (iteratedFDeriv Real 2 sourceTest z) <= C1, plus the intended Brownian coordinate unit-direction fact norm e <= 1, from the paper/testRegular bounded-Hessian interface. hSource remains explicit, and hSecondCoeff, Taylor moment decomposition, quadratic-variation normalization, and coordinate-sum leaves stay 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.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNorm
    2. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound
    3. hSecondFDerivOpNorm
    4. heUnit
    5. selected-test bounded-Hessian/regularity source hypothesis
    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.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound
    2. hFirst
    3. SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff
    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 cycle169GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDirectionalSecondDag :
    List ProofDagBlock :=
  [
    {
      id := "ASTIS.SALD.cycle169.middle_packet.line_second_from_ambient_directional_hessian"
      interface := "Compiled theorem: SALD.gaussianRealSelectedTestLineSecondDerivEqDirectionalFDeriv rewrites the global scalar-line iteratedDeriv 2 as the ambient selected-test second Frechet derivative applied twice to the affine-line direction e."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.gaussianRealSelectedTestLineSecondDerivEqDirectionalFDeriv",
        "hSource",
        "iteratedDeriv_vcomp_two",
        "iteratedDeriv_const_add",
        "iteratedDeriv_smul_const",
        "iteratedDeriv_fun_id",
        "deriv_smul_const",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "hLineSecond",
        "hNegLineSecond",
        "SALD.gaussianRealSelectedTestLineSecondBoundsOfDirectionalSecondBound"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle169.middle_packet.reflected_line_second_from_same_bound"
      interface := "Compiled theorem: SALD.gaussianRealSelectedTestLineSecondBoundsOfDirectionalSecondBound derives both hLineSecond and hNegLineSecond from one ambient diagonal directional-Hessian bound, using ContinuousMultilinearMap.map_smul_univ to cancel the two reflected signs."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.gaussianRealSelectedTestLineSecondBoundsOfDirectionalSecondBound",
        "SALD.gaussianRealSelectedTestLineSecondDerivEqDirectionalFDeriv",
        "hDirectionalSecond",
        "hLineSecond",
        "hNegLineSecond",
        "ContinuousMultilinearMap.map_smul_univ",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound",
        "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndLineSecondBounds"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle169.lower_2_packet.directional_hessian_from_second_fderiv_op_norm"
      interface := "Compiled theorem: SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNorm supplies hDirectionalSecond from a uniform selected-test second-Frechet-derivative operator-norm bound and the unit-direction side condition norm e <= 1."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle169GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDirectionalSecondLower2Obligation",
        "SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNorm",
        "hSecondFDerivOpNorm",
        "heUnit",
        "ContinuousMultilinearMap.le_mul_prod_of_opNorm_le_of_le",
        "Finset.prod_const",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "hDirectionalSecond",
        "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle169.middle_packet.directional_hessian_consumer"
      interface := "Compiled theorem: SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound feeds the ambient directional-Hessian bound into the accepted cycle-168 Taylor bridge and derives hC1 from the same norm bound."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle169GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDirectionalSecondMiddleObligation",
        "SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNorm",
        "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound",
        "SALD.gaussianRealSelectedTestLineSecondBoundsOfDirectionalSecondBound",
        "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndLineSecondBounds",
        "hDirectionalSecond",
        "hSource",
        "hC1",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "hFirst",
        "SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle169.remaining_second_fderiv_operator_norm_bound"
      interface := "Remaining exact theorem boundary after cycle 169 lower_2: expose the selected-test second-Frechet-derivative operator-norm bound forall z, norm (iteratedFDeriv Real 2 sourceTest z) <= C1, plus the intended Brownian coordinate unit-direction fact norm e <= 1, from the paper/testRegular bounded-Hessian interface. hSource remains explicit, and hSecondCoeff, Taylor moment decomposition, quadratic-variation normalization, and coordinate-sum leaves stay separate."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNorm",
        "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound",
        "hSecondFDerivOpNorm",
        "heUnit",
        "selected-test bounded-Hessian/regularity source hypothesis",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound",
        "hFirst",
        "SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff"
      ]
      status := ProofStatus.obligation
    }
  ]

/-! ### Cycle 170 lower_2: selected-test Hessian operator-norm source boundary -/

/-- Cycle-170 lower_2 narrowing from the Lean iterated-Frechet bound to the
source-facing Hessian operator-norm bound. -/

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