AutoSamplingTheory.SALD.cycle58UnifiedDiscreteGeneralMiddleContract

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 cycle58UnifiedDiscreteGeneralMiddleContract :
    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 58 middle: audit the upper unified/discrete general route in source order, verify that thm:unified-forward-KL remains only the c_t=u_t, m_t=w_t specialization of thm:general-moving-target-SALD, verify that thm:general-moving-target-SALD-discrete consumes the EM, frozen-delta, derivative/DV, LSI, and Gronwall interfaces, and select sald.general_moving_target_discrete.gronwall_side_conditions over appendix.tex:1573-1600 as the lower packet.

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 turn u_t+w_t into a transport velocity for pi_t.
2. main_body.tex:372-395 states thm:unified-forward-KL with the correction-field complexity E_alpha(pi_t,w_t) and the exact sigma_t^{-2} dot{s}(t)^{-1} coefficients.
3. appendix.tex:949-951 proves thm:unified-forward-KL only by specializing thm:general-moving-target-SALD with c_t=u_t and m_t=w_t.
4. appendix.tex:1313-1347 states thm:general-moving-target-SALD-discrete with sigma_eta factors, the doubled residual coefficient, Gamma/Delta terms, alpha ranges, and the constant inverse-schedule assumption.
5. appendix.tex:1354-1511 supplies the general EM endpoint/conditional-FP, frozen residual field split, and two sigma_eta^2/8 Young shares through existing obligations and scalar handoffs.
6. appendix.tex:1513-1570 applies eq:LSI-KL-FI and lem:dv_variation to the residual m_t under the EM interpolation law.
7. appendix.tex:1573-1583 defines K(t), stitches the EM endpoint laws, differentiates through s(t), and uses dot t(s(t))=dot s(t)^{-1}.
8. appendix.tex:1584-1597 gives the t-time Gronwall input with the theorem-display coefficients.
9. appendix.tex:1600 applies lem:gronwall; appendix.tex:1603 records the discrete guided VA-SALD specialization c_t=u_t.

leanStepMap : List String (explicit)

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

Ordered data items

1. Use SALD.cycle58UnifiedDiscreteGeneralUpperPacket and SALD.cycle58UnifiedDiscreteGeneralSkeletonObligation as parent route data; do not replace the cycle-57 continuous general route.
2. Route unified VA-SALD through SALD.unifiedForwardKlSpecializationContract, sald.unified_forward_kl.transport_velocity_bridge, sald.unified_forward_kl.specialization, SALD.generalVaSaldContract, and the cycle-57 guided/general middle and lower obligations.
3. Route appendix.tex:1313-1347 through SALD.generalMovingTargetDiscreteStatementContract and SALD.generalVaSaldDiscreteContract; both remain contractOnly.
4. Route appendix.tex:1354-1511 through SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, sald.general_moving_target_discrete.em_interpolation_fp, sald.general_moving_target_discrete.kl_derivative, cycle-53 derivative/DV scalar handoffs, and the frozen-delta obligations.
5. 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.
6. Route appendix.tex:1573-1600 through SALD.generalMovingTargetDiscreteGronwallInstantiationContract, SALD.generalMovingTargetDiscreteGronwallSideConditionContract, SALD.cycle20GeneralVaSaldDiscreteGronwallMiddleObligation, sald.general_moving_target_discrete.gronwall_application, and sald.general_moving_target_discrete.gronwall_side_conditions.
7. Select lower work exactly at SALD.generalMovingTargetDiscreteGronwallSideConditionContract / SALD.generalMovingTargetDiscreteGronwallSideConditionObligation / sald.general_moving_target_discrete.gronwall_side_conditions.
8. Keep local SLT one-step and disintegration material reference-only; this cycle performs no broad SLT import and marks no SLT theorem formalized.

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, 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 alpha-scaling witnesses.
3. eq:LSI-KL-FI remains the density, zero-set, admissible sqrt-density test, 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-57 derivative split, not reproved for the unified theorem.
5. The EM interpolation backend remains sald.general_moving_target_discrete.em_interpolation_fp; endpoint law and sigma-regrouping helpers compile only under explicit hypotheses and do not prove the conditional-FP backend.

obligations : List String (explicit)

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

Ordered data items

1. sald.unified_discrete_general.cycle58_upper_route
2. sald.unified_discrete_general.cycle58_middle_route_audit
3. sald.unified_forward_kl.transport_velocity_bridge
4. sald.unified_forward_kl.specialization
5. sald.general_moving_target.kl_derivative
6. sald.general_moving_target.gronwall_side_conditions
7. sald.general_moving_target_discrete.em_interpolation_fp
8. sald.general_moving_target_discrete.derivative_side_conditions
9. sald.general_moving_target_discrete.kl_derivative
10. sald.general_moving_target_discrete.dv_finite_log_mgf_witness
11. sald.general_moving_target_discrete.dv_m_energy
12. sald.general_moving_target_discrete.gronwall_application
13. sald.general_moving_target_discrete.gronwall_side_conditions
14. sald.general_moving_target_discrete.unified_specialization

lowerPacket : List String (explicit)

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

Ordered data items

1. Target exactly SALD.generalMovingTargetDiscreteGronwallSideConditionContract / SALD.generalMovingTargetDiscreteGronwallSideConditionObligation / sald.general_moving_target_discrete.gronwall_side_conditions.
2. First sub-slice: appendix.tex:1573-1583 endpoint stitching for K(t), including K(0)=KL(rho_0||pi_0), K(T)=KL(rho_K^eta||pi_T), interval compatibility, and constant-schedule admissibility.
3. Second sub-slice: appendix.tex:1584-1597 coefficient rewrites from eq:general_KL_derivative_8_discrete to the theorem display, reusing SALD.generalMovingTargetDiscreteConstantScheduleSquareScalar, SALD.generalMovingTargetDiscreteResidualCoefficientRewriteScalar, SALD.generalMovingTargetDiscreteGammaCoefficientRewriteScalar, and SALD.generalMovingTargetDiscreteDeltaCoefficientRewriteScalar.
4. Third sub-slice: appendix.tex:1600 Gronwall regularity and exact display matching for the source a(t) and b(t).
5. Do not restate either theorem, change coefficients, or promote Gronwall, DV, LSI/KL/FI, EM interpolation, KL derivative, frozen-delta, or endpoint stitching beyond the existing statuses.

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.cycle58UnifiedDiscreteGeneralMiddleObligation while remaining contractOnly.
2. SALD.generalVaSaldProofDag and SALD.generalVaSaldDiscreteProofDag contain ASTIS.SALD.unified_discrete_general.cycle58_middle_route_audit.
3. SALD.saldDependenciesForLabel entries for thm:unified-forward-KL and thm:general-moving-target-SALD-discrete include the cycle-58 middle contract, middle obligation, and route-audit node.
4. The lower packet is sald.general_moving_target_discrete.gronwall_side_conditions over appendix.tex:1573-1600, not a new KL derivative, DV, LSI, or EM proof.
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 cycle58UnifiedDiscreteGeneralMiddleContract :
    GeneralVaSaldGuidedPathMiddleContract where
  guidedResidualSource := saldGuidedResidualSource
  generalContinuousSource := saldGeneralMovingTargetSource
  unifiedSource := saldUnifiedForwardKlSource
  discreteSource := saldGeneralMovingTargetDiscreteSource
  objective := "Cycle 58 middle: audit the upper unified/discrete general route in source order, verify that thm:unified-forward-KL remains only the c_t=u_t, m_t=w_t specialization of thm:general-moving-target-SALD, verify that thm:general-moving-target-SALD-discrete consumes the EM, frozen-delta, derivative/DV, LSI, and Gronwall interfaces, and select sald.general_moving_target_discrete.gronwall_side_conditions over appendix.tex:1573-1600 as the lower packet."
  sourceStepMap := [
    "main_body.tex:359-368 uses prop:guided_path_residual and eq:poisson-eq to turn u_t+w_t into a transport velocity for pi_t.",
    "main_body.tex:372-395 states thm:unified-forward-KL with the correction-field complexity E_alpha(pi_t,w_t) and the exact sigma_t^{-2} dot{s}(t)^{-1} coefficients.",
    "appendix.tex:949-951 proves thm:unified-forward-KL only by specializing thm:general-moving-target-SALD with c_t=u_t and m_t=w_t.",
    "appendix.tex:1313-1347 states thm:general-moving-target-SALD-discrete with sigma_eta factors, the doubled residual coefficient, Gamma/Delta terms, alpha ranges, and the constant inverse-schedule assumption.",
    "appendix.tex:1354-1511 supplies the general EM endpoint/conditional-FP, frozen residual field split, and two sigma_eta^2/8 Young shares through existing obligations and scalar handoffs.",
    "appendix.tex:1513-1570 applies eq:LSI-KL-FI and lem:dv_variation to the residual m_t under the EM interpolation law.",
    "appendix.tex:1573-1583 defines K(t), stitches the EM endpoint laws, differentiates through s(t), and uses dot t(s(t))=dot s(t)^{-1}.",
    "appendix.tex:1584-1597 gives the t-time Gronwall input with the theorem-display coefficients.",
    "appendix.tex:1600 applies lem:gronwall; appendix.tex:1603 records the discrete guided VA-SALD specialization c_t=u_t."
  ]
  leanStepMap := [
    "Use SALD.cycle58UnifiedDiscreteGeneralUpperPacket and SALD.cycle58UnifiedDiscreteGeneralSkeletonObligation as parent route data; do not replace the cycle-57 continuous general route.",
    "Route unified VA-SALD through SALD.unifiedForwardKlSpecializationContract, sald.unified_forward_kl.transport_velocity_bridge, sald.unified_forward_kl.specialization, SALD.generalVaSaldContract, and the cycle-57 guided/general middle and lower obligations.",
    "Route appendix.tex:1313-1347 through SALD.generalMovingTargetDiscreteStatementContract and SALD.generalVaSaldDiscreteContract; both remain contractOnly.",
    "Route appendix.tex:1354-1511 through SALD.generalMovingTargetDiscreteDerivativeSideConditionContract, sald.general_moving_target_discrete.em_interpolation_fp, sald.general_moving_target_discrete.kl_derivative, cycle-53 derivative/DV scalar handoffs, and the frozen-delta obligations.",
    "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.generalMovingTargetDiscreteGronwallInstantiationContract, SALD.generalMovingTargetDiscreteGronwallSideConditionContract, SALD.cycle20GeneralVaSaldDiscreteGronwallMiddleObligation, sald.general_moving_target_discrete.gronwall_application, and sald.general_moving_target_discrete.gronwall_side_conditions.",
    "Select lower work exactly at SALD.generalMovingTargetDiscreteGronwallSideConditionContract / SALD.generalMovingTargetDiscreteGronwallSideConditionObligation / sald.general_moving_target_discrete.gronwall_side_conditions.",
    "Keep local SLT one-step and disintegration material reference-only; this cycle performs no broad SLT import and marks no SLT theorem formalized."
  ]
  citedResultInterfaces := [
    "lem:gronwall remains an endpoint-safe differentiability, FTC, 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 alpha-scaling witnesses.",
    "eq:LSI-KL-FI remains the density, zero-set, admissible sqrt-density test, 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-57 derivative split, not reproved for the unified theorem.",
    "The EM interpolation backend remains sald.general_moving_target_discrete.em_interpolation_fp; endpoint law and sigma-regrouping helpers compile only under explicit hypotheses and do not prove the conditional-FP backend."
  ]
  obligations := [
    "sald.unified_discrete_general.cycle58_upper_route",
    "sald.unified_discrete_general.cycle58_middle_route_audit",
    "sald.unified_forward_kl.transport_velocity_bridge",
    "sald.unified_forward_kl.specialization",
    "sald.general_moving_target.kl_derivative",
    "sald.general_moving_target.gronwall_side_conditions",
    "sald.general_moving_target_discrete.em_interpolation_fp",
    "sald.general_moving_target_discrete.derivative_side_conditions",
    "sald.general_moving_target_discrete.kl_derivative",
    "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",
    "sald.general_moving_target_discrete.unified_specialization"
  ]
  lowerPacket := [
    "Target exactly SALD.generalMovingTargetDiscreteGronwallSideConditionContract / SALD.generalMovingTargetDiscreteGronwallSideConditionObligation / sald.general_moving_target_discrete.gronwall_side_conditions.",
    "First sub-slice: appendix.tex:1573-1583 endpoint stitching for K(t), including K(0)=KL(rho_0||pi_0), K(T)=KL(rho_K^eta||pi_T), interval compatibility, and constant-schedule admissibility.",
    "Second sub-slice: appendix.tex:1584-1597 coefficient rewrites from eq:general_KL_derivative_8_discrete to the theorem display, reusing SALD.generalMovingTargetDiscreteConstantScheduleSquareScalar, SALD.generalMovingTargetDiscreteResidualCoefficientRewriteScalar, SALD.generalMovingTargetDiscreteGammaCoefficientRewriteScalar, and SALD.generalMovingTargetDiscreteDeltaCoefficientRewriteScalar.",
    "Third sub-slice: appendix.tex:1600 Gronwall regularity and exact display matching for the source a(t) and b(t).",
    "Do not restate either theorem, change coefficients, or promote Gronwall, DV, LSI/KL/FI, EM interpolation, KL derivative, frozen-delta, or endpoint stitching beyond the existing statuses."
  ]
  reviewerChecklist := [
    "SALD.unifiedForwardKlContract and SALD.generalVaSaldDiscreteContract list SALD.cycle58UnifiedDiscreteGeneralMiddleObligation while remaining contractOnly.",
    "SALD.generalVaSaldProofDag and SALD.generalVaSaldDiscreteProofDag contain ASTIS.SALD.unified_discrete_general.cycle58_middle_route_audit.",
    "SALD.saldDependenciesForLabel entries for thm:unified-forward-KL and thm:general-moving-target-SALD-discrete include the cycle-58 middle contract, middle obligation, and route-audit node.",
    "The lower packet is sald.general_moving_target_discrete.gronwall_side_conditions over appendix.tex:1573-1600, not a new KL derivative, DV, LSI, or EM proof.",
    "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-58 middle obligation tying the route audit to lower Gronwall/display
work. -/