Samplinglib
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AutoSamplingTheory.SALD.cycle58UnifiedDiscreteGeneralUpperPacket

Data definition / provenance and workflow record

Meaning and type

The result has data type AutoSamplingTheory.SALD.GeneralVaSaldUpperPacket. 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 cycle58UnifiedDiscreteGeneralUpperPacket : GeneralVaSaldUpperPacket

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.

objective : String (explicit)

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

Cycle 58 upper: no recovery is needed after the cycle-57 reviewer/build pass; Phase 1 is stable enough to re-close the unified and discrete general theorem route but not enough for broad cited-theory backfill; the single lower packet that reduces the largest remaining proof risk is sald.general_moving_target_discrete.gronwall_side_conditions over appendix.tex:1573-1600, using the cycle-53 derivative/DV handoff and cycle-56 Gronwall recovery pattern as inputs.
sourceLabels : List String (explicit)

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

Ordered data items

  1. thm:unified-forward-KL
  2. proof:thm:unified-forward-KL
  3. eq:residual-term
  4. eq:poisson-eq
  5. eq:SALD_Ito
  6. prop:guided_path_residual
  7. thm:general-moving-target-SALD
  8. thm:general-moving-target-SALD-discrete
  9. eq:general_moving_target_KL_bound_discrete
  10. eq:general_KL_derivative_7_discrete
  11. eq:general_KL_derivative_8_discrete
  12. eq:SALD_general_EM
  13. eq:general_moving_target_SALD_frozen_interp
  14. eq:general_discrete_delta_def
  15. lem:frozen_delta_cross_lip
  16. lem:gronwall
  17. lem:dv_variation
  18. eq:LSI-KL-FI
  19. def:alpha-complexity
modeDiscipline : List String (explicit)

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

Ordered data items

  1. Global phase judgment: cycle 57 passed reviewer and build gate, so there is no failed cycle to recover.
  2. Global phase judgment: Phase 1 theorem-skeleton translation is stable for a final unified/discrete general route refresh, but broad measure-theory or SDE backfill must still wait for this route to pass review.
  3. Global phase judgment: the largest remaining theorem-level proof risk is the discrete general Gronwall/display backend, because it stitches the EM intervals, applies the constant inverse-schedule rewrite, and matches the displayed theorem bound.
  4. Five-backend check 1, Gronwall: SALD.saldGronwallEndpointCalculusContract, SALD.generalMovingTargetDiscreteGronwallInstantiationContract, SALD.generalMovingTargetDiscreteGronwallSideConditionContract, and SALD.gronwallAnalyticObligation expose endpoint-safe differentiability, FTC/order integration, coefficient regularity, stitched endpoint laws, and display matching.
  5. Five-backend check 2, DV: dvVariationalFormulaInterface saldDvVariationSource, SALD.saldDvFiniteLogMgfContract, SALD.generalMovingTargetDvFiniteLogMgfWitnessContract, and SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract expose common-space, absolute-continuity, finite-KL, selected-test measurability, finite-log-mgf, and alpha-scaling witnesses.
  6. Five-backend check 3, LSI/KL/FI: SALD.saldLsiKlFiDensityTestContract and probability.lsi_to_kl_fi keep density, zero-set convention, admissible sqrt-density test, entropy identity, finite KL/FI, and Fisher chain-rule assumptions explicit.
  7. Five-backend check 4, continuous derivative: SALD.generalMovingTargetDerivativeCandidateContract, SALD.generalMovingTargetDerivativeObligation, and the cycle-57 derivative split keep the Fokker-Planck/KL derivative route source-cited or obligation-level for thm:general-moving-target-SALD and thm:unified-forward-KL.
  8. Five-backend check 5, EM interpolation: SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation, SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff, and sald.general_moving_target_discrete.em_interpolation_fp expose endpoint laws, conditional-law density, regular conditional drift, weak Fokker-Planck, and KL-differentiation side conditions without promoting the backend.
  9. FaithfulPaper Phase 1: use main_body.tex:359-395, appendix.tex:949-951, and appendix.tex:1313-1603; sald_version_2.tex remains excluded.
nonGoals : List String (explicit)

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

Ordered data items

  1. Do not restate thm:unified-forward-KL, thm:general-moving-target-SALD, or thm:general-moving-target-SALD-discrete.
  2. Do not promote SALD.unifiedForwardKlContract, SALD.generalVaSaldContract, SALD.generalVaSaldDiscreteContract, Gronwall, DV, LSI/KL/FI, continuous Fokker-Planck/KL differentiation, or EM interpolation above their existing contractOnly/sourceCited/obligation statuses.
  3. Do not replace the source route by a direct VA-SALD proof, Girsanov/path-space comparison, Pinsker/Talagrand/PI argument, or broad SLT import.
  4. Do not change the sigma_eta factors, doubled residual coefficient, Gamma/Delta terms, alpha ranges, constant inverse-schedule assumption, source labels, or theorem displays.
  5. Do not start systematic measure-theory or SDE backfill in this upper packet; if a backend is too large, sharpen the named source-cited interface or proof obligation.
lowerPacket : List String (explicit)

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

Ordered data items

  1. Middle should synchronize this cycle-58 upper route with conversion-windows/ASTIS-SALD-001.md, proof-obligations/ASTIS-SALD-001.md, SALD.generalVaSaldProofDag, and SALD.generalVaSaldDiscreteProofDag.
  2. Lower should target exactly SALD.generalMovingTargetDiscreteGronwallSideConditionContract / SALD.generalMovingTargetDiscreteGronwallSideConditionObligation / sald.general_moving_target_discrete.gronwall_side_conditions over appendix.tex:1573-1600.
  3. First lower sub-slice: endpoint and stitching interface for K(t)=KL(hat rho_{s(t)}||pi_t), including K(0)=KL(rho_0||pi_0), K(T)=KL(rho_K^eta||pi_T), interval compatibility, and constant schedule admissibility.
  4. Second lower sub-slice: coefficient rewrites from eq:general_KL_derivative_8_discrete to the theorem display, reusing SALD.generalMovingTargetDiscreteConstantScheduleSquareScalar, SALD.generalMovingTargetDiscreteResidualCoefficientRewriteScalar, SALD.generalMovingTargetDiscreteGammaCoefficientRewriteScalar, and SALD.generalMovingTargetDiscreteDeltaCoefficientRewriteScalar.
  5. Third lower sub-slice: Gronwall regularity/display matching for the exact a(t) and b(t), using the cycle-56 recovered Gronwall-accumulation route only as a pattern and not as a theorem substitution.
  6. If narrow backfill is attempted after the route check, keep it to one measure-theory detail around endpoint-law stitching or Measure.map-style law equality, guided by local SLT material as reference-only unless the imported theorem builds locally.
reviewerChecklist : List String (explicit)

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

Ordered data items

  1. SALD.unifiedForwardKlContract and SALD.generalVaSaldDiscreteContract list SALD.cycle58UnifiedDiscreteGeneralSkeletonObligation while remaining contractOnly.
  2. SALD.generalVaSaldProofDag and SALD.generalVaSaldDiscreteProofDag include ASTIS.SALD.unified_discrete_general.cycle58_upper_route and ASTIS.SALD.general_moving_target_discrete.cycle58_lower_packet.gronwall_display.
  3. SALD.saldDependenciesForLabel for thm:unified-forward-KL and thm:general-moving-target-SALD-discrete includes the cycle-58 packet, obligation, DAG, and lower-packet node.
  4. The five slow analytic interfaces remain ProofStatus.obligation or ProofStatus.sourceCited; no theorem contract or large analytic backend is marked formalized.
  5. The lower packet is theorem-level Gronwall/display stitching, not a new transcript expansion or isolated scalar sublemma.
  6. python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass.
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
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 cycle58UnifiedDiscreteGeneralUpperPacket : GeneralVaSaldUpperPacket where
  objective := "Cycle 58 upper: no recovery is needed after the cycle-57 reviewer/build pass; Phase 1 is stable enough to re-close the unified and discrete general theorem route but not enough for broad cited-theory backfill; the single lower packet that reduces the largest remaining proof risk is sald.general_moving_target_discrete.gronwall_side_conditions over appendix.tex:1573-1600, using the cycle-53 derivative/DV handoff and cycle-56 Gronwall recovery pattern as inputs."
  sourceLabels := [
    "thm:unified-forward-KL",
    "proof:thm:unified-forward-KL",
    "eq:residual-term",
    "eq:poisson-eq",
    "eq:SALD_Ito",
    "prop:guided_path_residual",
    "thm:general-moving-target-SALD",
    "thm:general-moving-target-SALD-discrete",
    "eq:general_moving_target_KL_bound_discrete",
    "eq:general_KL_derivative_7_discrete",
    "eq:general_KL_derivative_8_discrete",
    "eq:SALD_general_EM",
    "eq:general_moving_target_SALD_frozen_interp",
    "eq:general_discrete_delta_def",
    "lem:frozen_delta_cross_lip",
    "lem:gronwall",
    "lem:dv_variation",
    "eq:LSI-KL-FI",
    "def:alpha-complexity"
  ]
  modeDiscipline := [
    "Global phase judgment: cycle 57 passed reviewer and build gate, so there is no failed cycle to recover.",
    "Global phase judgment: Phase 1 theorem-skeleton translation is stable for a final unified/discrete general route refresh, but broad measure-theory or SDE backfill must still wait for this route to pass review.",
    "Global phase judgment: the largest remaining theorem-level proof risk is the discrete general Gronwall/display backend, because it stitches the EM intervals, applies the constant inverse-schedule rewrite, and matches the displayed theorem bound.",
    "Five-backend check 1, Gronwall: SALD.saldGronwallEndpointCalculusContract, SALD.generalMovingTargetDiscreteGronwallInstantiationContract, SALD.generalMovingTargetDiscreteGronwallSideConditionContract, and SALD.gronwallAnalyticObligation expose endpoint-safe differentiability, FTC/order integration, coefficient regularity, stitched endpoint laws, and display matching.",
    "Five-backend check 2, DV: dvVariationalFormulaInterface saldDvVariationSource, SALD.saldDvFiniteLogMgfContract, SALD.generalMovingTargetDvFiniteLogMgfWitnessContract, and SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract expose common-space, absolute-continuity, finite-KL, selected-test measurability, finite-log-mgf, and alpha-scaling witnesses.",
    "Five-backend check 3, LSI/KL/FI: SALD.saldLsiKlFiDensityTestContract and probability.lsi_to_kl_fi keep density, zero-set convention, admissible sqrt-density test, entropy identity, finite KL/FI, and Fisher chain-rule assumptions explicit.",
    "Five-backend check 4, continuous derivative: SALD.generalMovingTargetDerivativeCandidateContract, SALD.generalMovingTargetDerivativeObligation, and the cycle-57 derivative split keep the Fokker-Planck/KL derivative route source-cited or obligation-level for thm:general-moving-target-SALD and thm:unified-forward-KL.",
    "Five-backend check 5, EM interpolation: SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation, SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff, and sald.general_moving_target_discrete.em_interpolation_fp expose endpoint laws, conditional-law density, regular conditional drift, weak Fokker-Planck, and KL-differentiation side conditions without promoting the backend.",
    "FaithfulPaper Phase 1: use main_body.tex:359-395, appendix.tex:949-951, and appendix.tex:1313-1603; sald_version_2.tex remains excluded."
  ]
  nonGoals := [
    "Do not restate thm:unified-forward-KL, thm:general-moving-target-SALD, or thm:general-moving-target-SALD-discrete.",
    "Do not promote SALD.unifiedForwardKlContract, SALD.generalVaSaldContract, SALD.generalVaSaldDiscreteContract, Gronwall, DV, LSI/KL/FI, continuous Fokker-Planck/KL differentiation, or EM interpolation above their existing contractOnly/sourceCited/obligation statuses.",
    "Do not replace the source route by a direct VA-SALD proof, Girsanov/path-space comparison, Pinsker/Talagrand/PI argument, or broad SLT import.",
    "Do not change the sigma_eta factors, doubled residual coefficient, Gamma/Delta terms, alpha ranges, constant inverse-schedule assumption, source labels, or theorem displays.",
    "Do not start systematic measure-theory or SDE backfill in this upper packet; if a backend is too large, sharpen the named source-cited interface or proof obligation."
  ]
  lowerPacket := [
    "Middle should synchronize this cycle-58 upper route with conversion-windows/ASTIS-SALD-001.md, proof-obligations/ASTIS-SALD-001.md, SALD.generalVaSaldProofDag, and SALD.generalVaSaldDiscreteProofDag.",
    "Lower should target exactly SALD.generalMovingTargetDiscreteGronwallSideConditionContract / SALD.generalMovingTargetDiscreteGronwallSideConditionObligation / sald.general_moving_target_discrete.gronwall_side_conditions over appendix.tex:1573-1600.",
    "First lower sub-slice: endpoint and stitching interface for K(t)=KL(hat rho_{s(t)}||pi_t), including K(0)=KL(rho_0||pi_0), K(T)=KL(rho_K^eta||pi_T), interval compatibility, and constant schedule admissibility.",
    "Second lower sub-slice: coefficient rewrites from eq:general_KL_derivative_8_discrete to the theorem display, reusing SALD.generalMovingTargetDiscreteConstantScheduleSquareScalar, SALD.generalMovingTargetDiscreteResidualCoefficientRewriteScalar, SALD.generalMovingTargetDiscreteGammaCoefficientRewriteScalar, and SALD.generalMovingTargetDiscreteDeltaCoefficientRewriteScalar.",
    "Third lower sub-slice: Gronwall regularity/display matching for the exact a(t) and b(t), using the cycle-56 recovered Gronwall-accumulation route only as a pattern and not as a theorem substitution.",
    "If narrow backfill is attempted after the route check, keep it to one measure-theory detail around endpoint-law stitching or Measure.map-style law equality, guided by local SLT material as reference-only unless the imported theorem builds locally."
  ]
  reviewerChecklist := [
    "SALD.unifiedForwardKlContract and SALD.generalVaSaldDiscreteContract list SALD.cycle58UnifiedDiscreteGeneralSkeletonObligation while remaining contractOnly.",
    "SALD.generalVaSaldProofDag and SALD.generalVaSaldDiscreteProofDag include ASTIS.SALD.unified_discrete_general.cycle58_upper_route and ASTIS.SALD.general_moving_target_discrete.cycle58_lower_packet.gronwall_display.",
    "SALD.saldDependenciesForLabel for thm:unified-forward-KL and thm:general-moving-target-SALD-discrete includes the cycle-58 packet, obligation, DAG, and lower-packet node.",
    "The five slow analytic interfaces remain ProofStatus.obligation or ProofStatus.sourceCited; no theorem contract or large analytic backend is marked formalized.",
    "The lower packet is theorem-level Gronwall/display stitching, not a new transcript expansion or isolated scalar sublemma.",
    "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass."
  ]
  status := ProofStatus.obligation

/-- Cycle-58 obligation tying the final unified/discrete general theorem refresh
to explicit source-cited interfaces. -/

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