AutoSamplingTheory.SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianDag

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 cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianDag :
    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.cycle139.middle_packet.weak_fp_source_laplacian_field
interface : String (explicit)
Text describing intended mathematical interface.
Compiled dynamic-leaf theorem: SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField derives the weak-FP source-Laplacian pullback hweakFpLaplacianDef from hweakFpSourceActionDef and hweakFpSourceFieldEqLaplacian. This exposes the named source field behind the weak-FP abstract Laplacian action over appendix.tex:1379-1427 without using the circular hweakFpStdBasisDef route.
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.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianLowerObligation
2. SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField
3. Mathlib.Analysis.InnerProductSpace.Laplacian
4. appendix.tex:1379-1427
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. sald.general_moving_target_discrete.em_interpolation_fp
2. sald.discrete_forward_kl.em_interpolation_fp
3. thm:forward-KL-discrete
4. thm:general-moving-target-SALD-discrete
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.cycle139.middle_packet.test_laplacian_pointwise_weak_fp_source_field
interface : String (explicit)
Text describing intended mathematical interface.
Compiled downstream theorem: SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField feeds the weak-FP source-field split into SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfTestStdBasisSourceFormula. The test side still uses htestLaplacianStdBasisDef, while the weak-FP side uses hweakFpSourceActionDef and hweakFpSourceFieldEqLaplacian.
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.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianLowerObligation
2. SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField
3. SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField
4. SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfTestStdBasisSourceFormula
5. SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula
6. appendix.tex:1379-1427
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. sald.general_moving_target_discrete.em_interpolation_fp
2. sald.discrete_forward_kl.em_interpolation_fp
3. thm:forward-KL-discrete
4. thm:general-moving-target-SALD-discrete
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.cycle139.lower_1_packet.weak_fp_source_action_state_integral
interface : String (explicit)
Text describing intended mathematical interface.
Compiled lower_1 theorem: SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfStateIntegral derives hweakFpSourceActionDef from the EM state-law integral route. Inputs are hatRhoS = Measure.map hatXAtS P, sourceLaplacianFunctional as the hatRhoS integral, AEStronglyMeasurable weakFpLaplacianSourceField under hatRhoS, and the sample-space pullback integral formula for laplacian phi; Mathlib integral_map transports the sample integral to the named law.
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.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceActionIntegralScoutObligation
2. SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfStateIntegral
3. MeasureTheory.integral_map
4. appendix.tex:1379-1387
5. sald.general_moving_target_discrete.em_interpolation_fp
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. sald.general_moving_target_discrete.em_interpolation_fp
2. sald.discrete_forward_kl.em_interpolation_fp
3. thm:forward-KL-discrete
4. thm:general-moving-target-SALD-discrete
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.cycle139.lower_2_packet.source_laplacian_state_integral
interface : String (explicit)
Text describing intended mathematical interface.
Compiled lower_2 theorem: SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegral feeds the lower_1 state-law integral bridge after replacing the opaque weakFpLaplacianSourceField measurability and state-integral inputs by the concrete Mathlib source Laplacian field of selectedTest, using hweakFpSourceFieldEqLaplacian for the transport.
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.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianStateIntegralLowerObligation
2. SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegral
3. SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfStateIntegral
4. SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField
5. MeasureTheory.integral_map
6. Mathlib.Analysis.InnerProductSpace.Laplacian
7. appendix.tex:1379-1387
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. sald.general_moving_target_discrete.em_interpolation_fp
2. sald.discrete_forward_kl.em_interpolation_fp
3. thm:forward-KL-discrete
4. thm:general-moving-target-SALD-discrete
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.cycle139.remaining_weak_fp_source_laplacian_field
interface : String (explicit)
Text describing intended mathematical interface.
Remaining exact theorem boundary after cycle 139 lower_2: prove hweakFpSourceFieldEqLaplacian, sourceLaplacianFieldMeas for Laplacian.laplacian (selectedTest phi) under hatRhoS, and the source sample-space pullback integral formula hlaplacianSourceStateIntegral for that Laplacian field; keep the source functional definition, EM map-law measurability, htestLaplacianStdBasisDef, density-Laplacian source-action/source-field facts, second-Green residual/divergence, box-divergence, signed-face trace, pointwise trace equality, pointwise zero trace, and diffusion leaves explicit.
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.generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegral
2. SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfStateIntegral
3. SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField
4. SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField
5. SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfTestStdBasisSourceFormula
6. appendix.tex:1379-1427
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. sald.general_moving_target_discrete.em_interpolation_fp
2. sald.discrete_forward_kl.em_interpolation_fp
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.cycle139.reviewer_weak_fp_source_laplacian_check
interface : String (explicit)
Text describing intended mathematical interface.
Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hweakFpLaplacianDef is narrowed to hweakFpSourceActionDef plus hweakFpSourceFieldEqLaplacian through SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField, hweakFpSourceActionDef is further narrowed by SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegral to sourceLaplacianFieldMeas and hlaplacianSourceStateIntegral for the selected source Laplacian field, the downstream pointwise route uses SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField, htestLaplacianOperator is not reintroduced, hweakFpStdBasisDef and hweakFpDensityLaplacianAction are not used to close this weak-FP source definition, and python3 tools/astis.py check passes without SLT import, non-EM fallback, wrapper churn, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, 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.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianLowerObligation
2. SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceActionIntegralScoutObligation
3. SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianStateIntegralLowerObligation
4. SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegral
5. SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfStateIntegral
6. SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField
7. SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField
8. appendix.tex:1379-1427
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. cycle 139 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 cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianDag :
    List ProofDagBlock :=
  [
    {
      id := "ASTIS.SALD.cycle139.middle_packet.weak_fp_source_laplacian_field"
      interface := "Compiled dynamic-leaf theorem: SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField derives the weak-FP source-Laplacian pullback hweakFpLaplacianDef from hweakFpSourceActionDef and hweakFpSourceFieldEqLaplacian. This exposes the named source field behind the weak-FP abstract Laplacian action over appendix.tex:1379-1427 without using the circular hweakFpStdBasisDef route."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianLowerObligation",
        "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField",
        "Mathlib.Analysis.InnerProductSpace.Laplacian",
        "appendix.tex:1379-1427"
      ]
      reusedBy := [
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp",
        "thm:forward-KL-discrete",
        "thm:general-moving-target-SALD-discrete"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle139.middle_packet.test_laplacian_pointwise_weak_fp_source_field"
      interface := "Compiled downstream theorem: SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField feeds the weak-FP source-field split into SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfTestStdBasisSourceFormula. The test side still uses htestLaplacianStdBasisDef, while the weak-FP side uses hweakFpSourceActionDef and hweakFpSourceFieldEqLaplacian."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianLowerObligation",
        "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField",
        "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField",
        "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfTestStdBasisSourceFormula",
        "SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula",
        "appendix.tex:1379-1427"
      ]
      reusedBy := [
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp",
        "thm:forward-KL-discrete",
        "thm:general-moving-target-SALD-discrete"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle139.lower_1_packet.weak_fp_source_action_state_integral"
      interface := "Compiled lower_1 theorem: SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfStateIntegral derives hweakFpSourceActionDef from the EM state-law integral route. Inputs are hatRhoS = Measure.map hatXAtS P, sourceLaplacianFunctional as the hatRhoS integral, AEStronglyMeasurable weakFpLaplacianSourceField under hatRhoS, and the sample-space pullback integral formula for laplacian phi; Mathlib integral_map transports the sample integral to the named law."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceActionIntegralScoutObligation",
        "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfStateIntegral",
        "MeasureTheory.integral_map",
        "appendix.tex:1379-1387",
        "sald.general_moving_target_discrete.em_interpolation_fp"
      ]
      reusedBy := [
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp",
        "thm:forward-KL-discrete",
        "thm:general-moving-target-SALD-discrete"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle139.lower_2_packet.source_laplacian_state_integral"
      interface := "Compiled lower_2 theorem: SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegral feeds the lower_1 state-law integral bridge after replacing the opaque weakFpLaplacianSourceField measurability and state-integral inputs by the concrete Mathlib source Laplacian field of selectedTest, using hweakFpSourceFieldEqLaplacian for the transport."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianStateIntegralLowerObligation",
        "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegral",
        "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfStateIntegral",
        "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField",
        "MeasureTheory.integral_map",
        "Mathlib.Analysis.InnerProductSpace.Laplacian",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp",
        "thm:forward-KL-discrete",
        "thm:general-moving-target-SALD-discrete"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle139.remaining_weak_fp_source_laplacian_field"
      interface := "Remaining exact theorem boundary after cycle 139 lower_2: prove hweakFpSourceFieldEqLaplacian, sourceLaplacianFieldMeas for Laplacian.laplacian (selectedTest phi) under hatRhoS, and the source sample-space pullback integral formula hlaplacianSourceStateIntegral for that Laplacian field; keep the source functional definition, EM map-law measurability, htestLaplacianStdBasisDef, density-Laplacian source-action/source-field facts, second-Green residual/divergence, box-divergence, signed-face trace, pointwise trace equality, pointwise zero trace, and diffusion leaves explicit."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegral",
        "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfStateIntegral",
        "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField",
        "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField",
        "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfTestStdBasisSourceFormula",
        "appendix.tex:1379-1427"
      ]
      reusedBy := [
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp"
      ]
      status := ProofStatus.obligation
    },
    {
      id := "ASTIS.SALD.cycle139.reviewer_weak_fp_source_laplacian_check"
      interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hweakFpLaplacianDef is narrowed to hweakFpSourceActionDef plus hweakFpSourceFieldEqLaplacian through SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField, hweakFpSourceActionDef is further narrowed by SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegral to sourceLaplacianFieldMeas and hlaplacianSourceStateIntegral for the selected source Laplacian field, the downstream pointwise route uses SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField, htestLaplacianOperator is not reintroduced, hweakFpStdBasisDef and hweakFpDensityLaplacianAction are not used to close this weak-FP source definition, and python3 tools/astis.py check passes without SLT import, non-EM fallback, wrapper churn, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, or sald_version_2 use."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianLowerObligation",
        "SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceActionIntegralScoutObligation",
        "SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianStateIntegralLowerObligation",
        "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegral",
        "SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfStateIntegral",
        "SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField",
        "SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField",
        "appendix.tex:1379-1427"
      ]
      reusedBy := ["cycle 139 reviewer"]
      status := ProofStatus.obligation
    }
  ]

/-! ### Cycle 140: EM source-Laplacian state integral via frozen generator -/

/-- Cycle-140 narrowing of the source-Laplacian state-integral leaf to the
frozen EM generator component. -/

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