AutoSamplingTheory.SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalUpperPacket

Data definition / provenance and workflow record

Meaning and type

The result has data type AutoSamplingTheory.SALD.MainSkeletonAnalyticInterfaceLedger. A value of this type stores descriptions; it is not a proof of the statements in those descriptions.

def cycle81GeneralMovingTargetDiscreteEndpointConditionalUpperPacket :
    MainSkeletonAnalyticInterfaceLedger

Construction and field-by-field explanation

Construct a data record from explicit fields and the audited defaults shown below.

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.

sourceBlock : AutoSamplingTheory.SourceAnchor (explicit)

Nested provenance data.

AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteDerivativeSource — audited data reference, not expanded and not a compiled dependency edge

objective : String (explicit)

Descriptive text even when field names say formula, theorem, inequality, derivative, source gap, or proof.

Global phase judgment: cycle 80 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the single lower packet that best reduces the remaining risk is endpoint-law-to-conditional-law compatibility inside sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, specifically appendix.tex:1368-1377.

sourceLabels : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. proof:thm:general-moving-target-SALD-discrete:conditional-drift
2. proof:thm:general-moving-target-SALD-discrete:weak-conditional-fp
3. sald.general_moving_target_discrete.em_interpolation_fp
4. sald.general_moving_target_discrete.cycle81_endpoint_conditional_upper
5. thm:forward-KL-discrete
6. thm:general-moving-target-SALD-discrete

analyticInterfaces : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. Cycle 80 supplied-hypothesis drift-regularity packaging is accepted, but it did not construct a regular conditional law, condDistrib, condExpKernel, component conditional expectations, density/AC, or the weak FP theorem.
2. The active endpoint issue is still the Measure.map bookkeeping that identifies Law(hat X_s) as the marginal consumed by the conditional kernel for X_k^eta | hat X_s=x.
3. Cycles 71 and 76 contain local endpoint/marginal/orientation wrappers; cycles 74 and 75 contain the source-cited Mathlib conditional-kernel interfaces; cycle 81 asks lower to connect these pieces as the handoff consumed by cycle 80.
4. Weak FP source signs, KL differentiation, LSI/KL/FI, DV, Gronwall, and theorem closure remain downstream consumers and are not promoted.

theoremRoute : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. 1. Stay inside appendix.tex:1358-1387 with selected source slice appendix.tex:1368-1377.
2. 2. Preserve the paper definition bar b_{k,s}(x)=E[dot t_k*c_{t_k}(X_k^eta)+(sigma_eta^2/2)*nabla log pi_{t_k}(X_k^eta) | hat X_s=x].
3. 3. Reuse the existing endpoint law helpers SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation, SALD.generalMovingTargetDiscreteHatRhoMarginalOfJointMap, SALD.generalMovingTargetDiscreteEndpointMeasureMapToConditionalCompatibility, and the cycle-80 supplied drift-regularity wrapper.
4. 4. Lower should produce either a compiled supplied-hypothesis bridge from endpoint Measure.map facts to the conditional-law interface, such as SALD.generalMovingTargetDiscreteEndpointMeasureMapWeakFpPrereqHandoff, or one precise source-cited missing Mathlib conditional-law theorem; it should not advance to weak FP or KL derivative work in this packet.

modeDiscipline : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. faithfulPaper Phase 1 only: preserve the original appendix.tex route and keep sald_version_2.tex out of scope.
2. Do not add standard-Borel, probability, marginal-compatibility, conditional-law, density, or test-regularity assumptions to theorem statements; keep them in source-cited interfaces or obligations.
3. Use lean-stat-learning-theory only as a local Mathlib style reference; do not import it, change Lake dependencies, or claim an SLT theorem is formalized.

nonGoals : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. No broad theorem-route audit, source-index rebaseline beyond the gate, display algebra, Gronwall/DV/LSI work, frozen-delta work, broad reusable API design, or project-article export.
2. No proof of the weak conditional Fokker-Planck theorem, generator theorem, KL derivative theorem, or theorem closure.
3. No status promotion for the EM interpolation FP backend, conditional law construction, weak FP, KL derivative, or either discrete theorem contract.

lowerPacket : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. Target exactly sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, narrowed to endpoint-law-to-conditional-law compatibility for appendix.tex:1368-1377.
2. Middle must keep two-way synchronization among the source line, Lean declarations, proof-obligation rows, and any run handoff; it should not open an unrelated theorem-route audit.
3. Lower should connect the endpoint Measure.map laws and named hat rho_s marginal to the conditional kernel orientation required by the existing conditional-law/measurability interface and cycle-80 drift-regularity handoff.
4. Cycle 81 lower compiles SALD.generalMovingTargetDiscreteEndpointMeasureMapWeakFpPrereqHandoff as the endpoint-only bridge once bar b_{k,s} measurability/integrability and the weak-FP prerequisite consumer are supplied.
5. If blocked, record one exact Mathlib conditional-distribution or conditional-expectation theorem boundary below formalized status, with marginal compatibility, measurability, and vector-valued integrability hypotheses explicit.

reviewerChecklist : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. The active lower packet still targets sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, specifically appendix.tex:1368-1377.
2. Cycle 81 uses existing cycle-71/cycle-76 endpoint wrappers, cycle-74/cycle-75 conditional-kernel interfaces, and cycle-80 drift-regularity handoff instead of duplicating or promoting them.
3. Both discrete theorem contracts remain contractOnly; conditional law, weak FP, generator-to-law, KL derivative, density/AC, LSI/KL/FI, DV, and Gronwall remain sourceCited or obligation unless a local declaration compiles.
4. SLT remains reference-only with no Lake dependency or import change.
5. python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass.

dependencies : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract
2. SALD.generalMovingTargetDiscreteHatRhoMarginalOfJointMap
3. SALD.generalMovingTargetDiscreteEndpointMeasureMapToConditionalCompatibility
4. SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalLowerObligation
5. SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalLowerObligation
6. SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface
7. SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation
8. SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation
9. SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff
10. SALD.generalMovingTargetDiscreteEndpointMeasureMapWeakFpPrereqHandoff
11. sald.general_moving_target_discrete.em_interpolation_fp

status : AutoSamplingTheory.ProofStatus (explicit)

Stored workflow tag; honor the exact default but do not infer mathematical certification.

AutoSamplingTheory.ProofStatus.obligation — stored label only; no proof certification

def cycle81GeneralMovingTargetDiscreteEndpointConditionalUpperPacket :
    MainSkeletonAnalyticInterfaceLedger where
  sourceBlock := saldGeneralMovingTargetDiscreteDerivativeSource
  objective := "Global phase judgment: cycle 80 passed reviewer/build and needs no recovery; Phase 1 theorem-skeleton translation is stable enough for cited-theory backfill; the single lower packet that best reduces the remaining risk is endpoint-law-to-conditional-law compatibility inside sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, specifically appendix.tex:1368-1377."
  sourceLabels := [
    "proof:thm:general-moving-target-SALD-discrete:conditional-drift",
    "proof:thm:general-moving-target-SALD-discrete:weak-conditional-fp",
    "sald.general_moving_target_discrete.em_interpolation_fp",
    "sald.general_moving_target_discrete.cycle81_endpoint_conditional_upper",
    "thm:forward-KL-discrete",
    "thm:general-moving-target-SALD-discrete"
  ]
  analyticInterfaces := [
    "Cycle 80 supplied-hypothesis drift-regularity packaging is accepted, but it did not construct a regular conditional law, condDistrib, condExpKernel, component conditional expectations, density/AC, or the weak FP theorem.",
    "The active endpoint issue is still the Measure.map bookkeeping that identifies Law(hat X_s) as the marginal consumed by the conditional kernel for X_k^eta | hat X_s=x.",
    "Cycles 71 and 76 contain local endpoint/marginal/orientation wrappers; cycles 74 and 75 contain the source-cited Mathlib conditional-kernel interfaces; cycle 81 asks lower to connect these pieces as the handoff consumed by cycle 80.",
    "Weak FP source signs, KL differentiation, LSI/KL/FI, DV, Gronwall, and theorem closure remain downstream consumers and are not promoted."
  ]
  theoremRoute := [
    "1. Stay inside appendix.tex:1358-1387 with selected source slice appendix.tex:1368-1377.",
    "2. Preserve the paper definition bar b_{k,s}(x)=E[dot t_k*c_{t_k}(X_k^eta)+(sigma_eta^2/2)*nabla log pi_{t_k}(X_k^eta) | hat X_s=x].",
    "3. Reuse the existing endpoint law helpers SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation, SALD.generalMovingTargetDiscreteHatRhoMarginalOfJointMap, SALD.generalMovingTargetDiscreteEndpointMeasureMapToConditionalCompatibility, and the cycle-80 supplied drift-regularity wrapper.",
    "4. Lower should produce either a compiled supplied-hypothesis bridge from endpoint Measure.map facts to the conditional-law interface, such as SALD.generalMovingTargetDiscreteEndpointMeasureMapWeakFpPrereqHandoff, or one precise source-cited missing Mathlib conditional-law theorem; it should not advance to weak FP or KL derivative work in this packet."
  ]
  modeDiscipline := [
    "faithfulPaper Phase 1 only: preserve the original appendix.tex route and keep sald_version_2.tex out of scope.",
    "Do not add standard-Borel, probability, marginal-compatibility, conditional-law, density, or test-regularity assumptions to theorem statements; keep them in source-cited interfaces or obligations.",
    "Use lean-stat-learning-theory only as a local Mathlib style reference; do not import it, change Lake dependencies, or claim an SLT theorem is formalized."
  ]
  nonGoals := [
    "No broad theorem-route audit, source-index rebaseline beyond the gate, display algebra, Gronwall/DV/LSI work, frozen-delta work, broad reusable API design, or project-article export.",
    "No proof of the weak conditional Fokker-Planck theorem, generator theorem, KL derivative theorem, or theorem closure.",
    "No status promotion for the EM interpolation FP backend, conditional law construction, weak FP, KL derivative, or either discrete theorem contract."
  ]
  lowerPacket := [
    "Target exactly sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, narrowed to endpoint-law-to-conditional-law compatibility for appendix.tex:1368-1377.",
    "Middle must keep two-way synchronization among the source line, Lean declarations, proof-obligation rows, and any run handoff; it should not open an unrelated theorem-route audit.",
    "Lower should connect the endpoint Measure.map laws and named hat rho_s marginal to the conditional kernel orientation required by the existing conditional-law/measurability interface and cycle-80 drift-regularity handoff.",
    "Cycle 81 lower compiles SALD.generalMovingTargetDiscreteEndpointMeasureMapWeakFpPrereqHandoff as the endpoint-only bridge once bar b_{k,s} measurability/integrability and the weak-FP prerequisite consumer are supplied.",
    "If blocked, record one exact Mathlib conditional-distribution or conditional-expectation theorem boundary below formalized status, with marginal compatibility, measurability, and vector-valued integrability hypotheses explicit."
  ]
  reviewerChecklist := [
    "The active lower packet still targets sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex:1358-1387, specifically appendix.tex:1368-1377.",
    "Cycle 81 uses existing cycle-71/cycle-76 endpoint wrappers, cycle-74/cycle-75 conditional-kernel interfaces, and cycle-80 drift-regularity handoff instead of duplicating or promoting them.",
    "Both discrete theorem contracts remain contractOnly; conditional law, weak FP, generator-to-law, KL derivative, density/AC, LSI/KL/FI, DV, and Gronwall remain sourceCited or obligation unless a local declaration compiles.",
    "SLT remains reference-only with no Lake dependency or import change.",
    "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass."
  ]
  dependencies := [
    "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract",
    "SALD.generalMovingTargetDiscreteHatRhoMarginalOfJointMap",
    "SALD.generalMovingTargetDiscreteEndpointMeasureMapToConditionalCompatibility",
    "SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalLowerObligation",
    "SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalLowerObligation",
    "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface",
    "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation",
    "SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation",
    "SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff",
    "SALD.generalMovingTargetDiscreteEndpointMeasureMapWeakFpPrereqHandoff",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  status := ProofStatus.obligation

/-- Cycle-81 upper obligation for the endpoint-to-conditional packet. -/