AutoSamplingTheory.SALD.cycle68UnifiedDiscreteGeneralSkeletonMiddleContract

Data definition / provenance and workflow record

Meaning and type

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

def cycle68UnifiedDiscreteGeneralSkeletonMiddleContract :
    GeneralVaSaldGuidedPathMiddleContract

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.

guidedResidualSource : AutoSamplingTheory.SourceAnchor (explicit)

Nested provenance data.

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

generalContinuousSource : AutoSamplingTheory.SourceAnchor (explicit)

Nested provenance data.

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

unifiedSource : AutoSamplingTheory.SourceAnchor (explicit)

Nested provenance data.

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

discreteSource : AutoSamplingTheory.SourceAnchor (explicit)

Nested provenance data.

AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteSource — 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.

Cycle 68 middle: audit the upper unified/discrete-general route after the accepted cycle-67 guided/general pass, verify thm:unified-forward-KL remains only the appendix specialization of thm:general-moving-target-SALD with c_t=u_t and m_t=w_t, verify thm:general-moving-target-SALD-discrete consumes the general EM endpoint/conditional-FP, frozen-delta, LSI, residual-DV, time-change, and Gronwall/display interfaces in source order, and keep lower work on SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeObligation / sald.unified_discrete_general.cycle68_discrete_general_bridge.

sourceStepMap : List String (explicit)

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

Ordered data items

1. main_body.tex:359-368 uses prop:guided_path_residual and eq:poisson-eq to make u_t+w_t a transport velocity for pi_t.
2. main_body.tex:372-395 states thm:unified-forward-KL with correction-field complexity E_alpha(pi_t,w_t) and the exact sigma_t^(-2)*dot{s}(t)^(-1) coefficient.
3. appendix.tex:949-951 proves thm:unified-forward-KL only by specializing thm:general-moving-target-SALD with c_t<-u_t.
4. appendix.tex:1313-1347 states thm:general-moving-target-SALD-discrete with the doubled residual coefficient, sigma_eta factors, Gamma/Delta terms, alpha ranges, and constant inverse-schedule assumption.
5. appendix.tex:1354-1387 fixes the EM interpolation endpoint laws, differentiates KL(hat rho_s||tilde pi_s), defines the frozen conditional drift, and invokes the conditional Fokker-Planck equation.
6. appendix.tex:1389-1511 splits the Laplacian, identifies delta_pi^VA+dot t(s)*m_t, applies the frozen/residual Young bounds, and consumes lem:frozen_delta_cross_lip.
7. appendix.tex:1513-1570 applies eq:LSI-KL-FI and lem:dv_variation to the residual field m_t under the EM interpolation law.
8. appendix.tex:1573-1600 changes variables to K(t), uses dot t(s(t))=dot s(t)^(-1), forms the pointwise Gronwall input, and applies lem:gronwall to match the theorem display.
9. appendix.tex:1603 records the downstream discrete guided VA-SALD specialization c_t=u_t; this middle audit does not make it a direct proof of thm:unified-forward-KL.

leanStepMap : List String (explicit)

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

Ordered data items

1. Use SALD.cycle68UnifiedDiscreteGeneralSkeletonUpperPacket and SALD.cycle68UnifiedDiscreteGeneralSkeletonObligation as parent route data; keep the cycle-67 continuous guided/general route as the source of thm:unified-forward-KL.
2. Route thm:unified-forward-KL through SALD.guidedResidualIdentityContract, SALD.unifiedForwardKlSpecializationContract, sald.unified_forward_kl.transport_velocity_bridge, sald.unified_forward_kl.specialization, SALD.generalVaSaldContract, and the cycle-67 middle and lower obligations.
3. Route appendix.tex:1313-1347 through SALD.generalMovingTargetDiscreteStatementContract and SALD.generalVaSaldDiscreteContract; both remain contractOnly.
4. Route appendix.tex:1354-1387 through SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation, the cycle-63 endpoint Measure.map helpers, SALD.generalMovingTargetDiscreteConditionalDriftContract, SALD.cycle64GeneralMovingTargetDiscreteConditionalDriftLowerObligation, and sald.general_moving_target_discrete.em_interpolation_fp.
5. Route appendix.tex:1389-1511 through SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, SALD.generalMovingTargetDiscreteFrozenResidualAlgebraVector, the cycle-28 Young/Fisher scalar helpers, lem:frozen_delta_cross_lip, and sald.general_moving_target_discrete.kl_derivative.
6. Route appendix.tex:1513-1570 through SALD.saldLsiKlFiDensityTestContract, probability.lsi_to_kl_fi, dvVariationalFormulaInterface saldDvVariationSource, SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract, and sald.general_moving_target_discrete.dv_m_energy.
7. Route appendix.tex:1573-1600 through SALD.generalMovingTargetDiscreteDerivativeDvTimeChangedScalar, SALD.generalMovingTargetDiscretePointwiseGronwallInputOfPostDvTimeChanged, SALD.generalMovingTargetDiscreteGronwallInstantiationContract, SALD.generalMovingTargetDiscreteGronwallSideConditionContract, SALD.generalMovingTargetDiscreteGronwallNamedCoefficientInput, and SALD.generalMovingTargetDiscreteGronwallEndpointRewriteScalar.
8. Select lower work exactly at SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeObligation / sald.unified_discrete_general.cycle68_discrete_general_bridge; sharpen that source-cited bridge if blocked, rather than changing theorem statements or backend statuses.

citedResultInterfaces : List String (explicit)

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

Ordered data items

1. lem:gronwall remains an endpoint-safe differentiability, FTC/order integration, coefficient-regularity, endpoint-stitching, and display-matching obligation.
2. lem:dv_variation remains source-cited through common-space, absolute-continuity, finite-KL, selected-test measurability, finite-log-mgf, and positive-alpha witnesses.
3. eq:LSI-KL-FI remains the density, zero-set convention, admissible sqrt-density test or approximation, entropy identity, finite KL/FI, and Fisher-chain obligation.
4. The continuous Fokker-Planck/KL derivative is consumed upstream through sald.general_moving_target.kl_derivative and the cycle-67 residual-to-Gronwall bridge; it is not reproved for the unified theorem.
5. The EM interpolation backend remains sald.general_moving_target_discrete.em_interpolation_fp; cycle-63 endpoint helpers and the cycle-64 conditional-drift algebra do not prove conditional laws, density/AC, weak Fokker-Planck, or KL differentiation.
6. Local SLT one-step/disintegration material remains reference-only for later narrow backfill; this middle packet imports no SLT theorem and promotes no analytic backend.

obligations : List String (explicit)

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

Ordered data items

1. sald.unified_discrete_general.cycle68_upper_route
2. sald.unified_discrete_general.cycle68_middle_route_audit
3. sald.unified_discrete_general.cycle68_discrete_general_bridge
4. sald.guided_general.cycle67_middle_route_audit
5. sald.general_moving_target.cycle67_residual_to_gronwall_bridge
6. sald.general_moving_target.cycle67_residual_to_gronwall_lower
7. sald.unified_forward_kl.transport_velocity_bridge
8. sald.unified_forward_kl.specialization
9. sald.general_moving_target_discrete.cycle48_em_endpoint_conditional_fp_audit
10. sald.general_moving_target_discrete.em_interpolation_fp
11. sald.general_moving_target_discrete.frozen_delta_cross_lip
12. sald.general_moving_target_discrete.kl_derivative
13. probability.lsi_to_kl_fi
14. sald.general_moving_target_discrete.dv_finite_log_mgf_witness
15. sald.general_moving_target_discrete.dv_m_energy
16. sald.general_moving_target_discrete.gronwall_application
17. sald.general_moving_target_discrete.gronwall_side_conditions

lowerPacket : List String (explicit)

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

Ordered data items

1. Target exactly SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeObligation / sald.unified_discrete_general.cycle68_discrete_general_bridge.
2. First sub-slice: verify main_body.tex:359-395 and appendix.tex:949-951 against the guided residual, correction-field transport bridge, and cycle-67 continuous general theorem route.
3. Second sub-slice: verify appendix.tex:1313-1387 against the fixed discrete statement, endpoint-law/common-space helpers, conditional drift, weak EM Fokker-Planck, and KL derivative side-condition interfaces.
4. Third sub-slice: verify appendix.tex:1389-1570 against frozen-delta, Young/LSI, residual DV with Z=alpha*||m_t||^2, and the post-DV time-change handoff.
5. Fourth sub-slice: verify appendix.tex:1573-1600 and theorem display lines 1316-1347 against the Gronwall side-condition and display-matching obligations.
6. Do not restate either theorem, change coefficients, or promote Gronwall, DV, LSI/KL/FI, EM interpolation, KL derivative, frozen-delta, endpoint stitching, or theorem status.

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.cycle68UnifiedDiscreteGeneralSkeletonMiddleObligation while remaining contractOnly.
2. SALD.generalVaSaldProofDag and SALD.generalVaSaldDiscreteProofDag contain ASTIS.SALD.unified_discrete_general.cycle68_middle_route_audit between the upper route nodes and the selected lower bridge.
3. SALD.saldDependenciesForLabel entries for thm:unified-forward-KL and thm:general-moving-target-SALD-discrete include the cycle-68 middle contract, middle obligation, and route-audit node.
4. The selected lower packet remains sald.unified_discrete_general.cycle68_discrete_general_bridge and no broad SLT/SDE backfill is started before reviewer acceptance.
5. No theorem statement, source constant, source label, source-file scope, SLT reuse status, or analytic backend status changes.
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

def cycle68UnifiedDiscreteGeneralSkeletonMiddleContract :
    GeneralVaSaldGuidedPathMiddleContract where
  guidedResidualSource := saldGuidedResidualSource
  generalContinuousSource := saldGeneralMovingTargetSource
  unifiedSource := saldUnifiedForwardKlSource
  discreteSource := saldGeneralMovingTargetDiscreteSource
  objective := "Cycle 68 middle: audit the upper unified/discrete-general route after the accepted cycle-67 guided/general pass, verify thm:unified-forward-KL remains only the appendix specialization of thm:general-moving-target-SALD with c_t=u_t and m_t=w_t, verify thm:general-moving-target-SALD-discrete consumes the general EM endpoint/conditional-FP, frozen-delta, LSI, residual-DV, time-change, and Gronwall/display interfaces in source order, and keep lower work on SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeObligation / sald.unified_discrete_general.cycle68_discrete_general_bridge."
  sourceStepMap := [
    "main_body.tex:359-368 uses prop:guided_path_residual and eq:poisson-eq to make u_t+w_t a transport velocity for pi_t.",
    "main_body.tex:372-395 states thm:unified-forward-KL with correction-field complexity E_alpha(pi_t,w_t) and the exact sigma_t^(-2)*dot{s}(t)^(-1) coefficient.",
    "appendix.tex:949-951 proves thm:unified-forward-KL only by specializing thm:general-moving-target-SALD with c_t<-u_t.",
    "appendix.tex:1313-1347 states thm:general-moving-target-SALD-discrete with the doubled residual coefficient, sigma_eta factors, Gamma/Delta terms, alpha ranges, and constant inverse-schedule assumption.",
    "appendix.tex:1354-1387 fixes the EM interpolation endpoint laws, differentiates KL(hat rho_s||tilde pi_s), defines the frozen conditional drift, and invokes the conditional Fokker-Planck equation.",
    "appendix.tex:1389-1511 splits the Laplacian, identifies delta_pi^VA+dot t(s)*m_t, applies the frozen/residual Young bounds, and consumes lem:frozen_delta_cross_lip.",
    "appendix.tex:1513-1570 applies eq:LSI-KL-FI and lem:dv_variation to the residual field m_t under the EM interpolation law.",
    "appendix.tex:1573-1600 changes variables to K(t), uses dot t(s(t))=dot s(t)^(-1), forms the pointwise Gronwall input, and applies lem:gronwall to match the theorem display.",
    "appendix.tex:1603 records the downstream discrete guided VA-SALD specialization c_t=u_t; this middle audit does not make it a direct proof of thm:unified-forward-KL."
  ]
  leanStepMap := [
    "Use SALD.cycle68UnifiedDiscreteGeneralSkeletonUpperPacket and SALD.cycle68UnifiedDiscreteGeneralSkeletonObligation as parent route data; keep the cycle-67 continuous guided/general route as the source of thm:unified-forward-KL.",
    "Route thm:unified-forward-KL through SALD.guidedResidualIdentityContract, SALD.unifiedForwardKlSpecializationContract, sald.unified_forward_kl.transport_velocity_bridge, sald.unified_forward_kl.specialization, SALD.generalVaSaldContract, and the cycle-67 middle and lower obligations.",
    "Route appendix.tex:1313-1347 through SALD.generalMovingTargetDiscreteStatementContract and SALD.generalVaSaldDiscreteContract; both remain contractOnly.",
    "Route appendix.tex:1354-1387 through SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation, the cycle-63 endpoint Measure.map helpers, SALD.generalMovingTargetDiscreteConditionalDriftContract, SALD.cycle64GeneralMovingTargetDiscreteConditionalDriftLowerObligation, and sald.general_moving_target_discrete.em_interpolation_fp.",
    "Route appendix.tex:1389-1511 through SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, SALD.generalMovingTargetDiscreteFrozenResidualAlgebraVector, the cycle-28 Young/Fisher scalar helpers, lem:frozen_delta_cross_lip, and sald.general_moving_target_discrete.kl_derivative.",
    "Route appendix.tex:1513-1570 through SALD.saldLsiKlFiDensityTestContract, probability.lsi_to_kl_fi, dvVariationalFormulaInterface saldDvVariationSource, SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract, and sald.general_moving_target_discrete.dv_m_energy.",
    "Route appendix.tex:1573-1600 through SALD.generalMovingTargetDiscreteDerivativeDvTimeChangedScalar, SALD.generalMovingTargetDiscretePointwiseGronwallInputOfPostDvTimeChanged, SALD.generalMovingTargetDiscreteGronwallInstantiationContract, SALD.generalMovingTargetDiscreteGronwallSideConditionContract, SALD.generalMovingTargetDiscreteGronwallNamedCoefficientInput, and SALD.generalMovingTargetDiscreteGronwallEndpointRewriteScalar.",
    "Select lower work exactly at SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeObligation / sald.unified_discrete_general.cycle68_discrete_general_bridge; sharpen that source-cited bridge if blocked, rather than changing theorem statements or backend statuses."
  ]
  citedResultInterfaces := [
    "lem:gronwall remains an endpoint-safe differentiability, FTC/order integration, coefficient-regularity, endpoint-stitching, and display-matching obligation.",
    "lem:dv_variation remains source-cited through common-space, absolute-continuity, finite-KL, selected-test measurability, finite-log-mgf, and positive-alpha witnesses.",
    "eq:LSI-KL-FI remains the density, zero-set convention, admissible sqrt-density test or approximation, entropy identity, finite KL/FI, and Fisher-chain obligation.",
    "The continuous Fokker-Planck/KL derivative is consumed upstream through sald.general_moving_target.kl_derivative and the cycle-67 residual-to-Gronwall bridge; it is not reproved for the unified theorem.",
    "The EM interpolation backend remains sald.general_moving_target_discrete.em_interpolation_fp; cycle-63 endpoint helpers and the cycle-64 conditional-drift algebra do not prove conditional laws, density/AC, weak Fokker-Planck, or KL differentiation.",
    "Local SLT one-step/disintegration material remains reference-only for later narrow backfill; this middle packet imports no SLT theorem and promotes no analytic backend."
  ]
  obligations := [
    "sald.unified_discrete_general.cycle68_upper_route",
    "sald.unified_discrete_general.cycle68_middle_route_audit",
    "sald.unified_discrete_general.cycle68_discrete_general_bridge",
    "sald.guided_general.cycle67_middle_route_audit",
    "sald.general_moving_target.cycle67_residual_to_gronwall_bridge",
    "sald.general_moving_target.cycle67_residual_to_gronwall_lower",
    "sald.unified_forward_kl.transport_velocity_bridge",
    "sald.unified_forward_kl.specialization",
    "sald.general_moving_target_discrete.cycle48_em_endpoint_conditional_fp_audit",
    "sald.general_moving_target_discrete.em_interpolation_fp",
    "sald.general_moving_target_discrete.frozen_delta_cross_lip",
    "sald.general_moving_target_discrete.kl_derivative",
    "probability.lsi_to_kl_fi",
    "sald.general_moving_target_discrete.dv_finite_log_mgf_witness",
    "sald.general_moving_target_discrete.dv_m_energy",
    "sald.general_moving_target_discrete.gronwall_application",
    "sald.general_moving_target_discrete.gronwall_side_conditions"
  ]
  lowerPacket := [
    "Target exactly SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeObligation / sald.unified_discrete_general.cycle68_discrete_general_bridge.",
    "First sub-slice: verify main_body.tex:359-395 and appendix.tex:949-951 against the guided residual, correction-field transport bridge, and cycle-67 continuous general theorem route.",
    "Second sub-slice: verify appendix.tex:1313-1387 against the fixed discrete statement, endpoint-law/common-space helpers, conditional drift, weak EM Fokker-Planck, and KL derivative side-condition interfaces.",
    "Third sub-slice: verify appendix.tex:1389-1570 against frozen-delta, Young/LSI, residual DV with Z=alpha*||m_t||^2, and the post-DV time-change handoff.",
    "Fourth sub-slice: verify appendix.tex:1573-1600 and theorem display lines 1316-1347 against the Gronwall side-condition and display-matching obligations.",
    "Do not restate either theorem, change coefficients, or promote Gronwall, DV, LSI/KL/FI, EM interpolation, KL derivative, frozen-delta, endpoint stitching, or theorem status."
  ]
  reviewerChecklist := [
    "SALD.unifiedForwardKlContract and SALD.generalVaSaldDiscreteContract list SALD.cycle68UnifiedDiscreteGeneralSkeletonMiddleObligation while remaining contractOnly.",
    "SALD.generalVaSaldProofDag and SALD.generalVaSaldDiscreteProofDag contain ASTIS.SALD.unified_discrete_general.cycle68_middle_route_audit between the upper route nodes and the selected lower bridge.",
    "SALD.saldDependenciesForLabel entries for thm:unified-forward-KL and thm:general-moving-target-SALD-discrete include the cycle-68 middle contract, middle obligation, and route-audit node.",
    "The selected lower packet remains sald.unified_discrete_general.cycle68_discrete_general_bridge and no broad SLT/SDE backfill is started before reviewer acceptance.",
    "No theorem statement, source constant, source label, source-file scope, SLT reuse status, or analytic backend status changes.",
    "python3 tools/astis.py source-index ASTIS-SALD-001 and python3 tools/astis.py check pass."
  ]
  status := ProofStatus.obligation

/-- Cycle-68 middle obligation tying the route audit to the selected
unified/discrete general source-cited bridge. -/