AutoSamplingTheory.SALD.cycle20GeneralVaSaldMiddleContract

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 cycle20GeneralVaSaldMiddleContract :
    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.

Translate the cycle-20 discrete general Gronwall/display bridge into a lower-ready map for sald.general_moving_target_discrete.gronwall_side_conditions, with endpoint stitching, constant-schedule coefficient rewrites, coefficient regularity, and theorem-display matching kept as explicit obligations.

sourceStepMap : List String (explicit)

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

Ordered data items

1. appendix.tex:1316-1347 fixes the theorem display for KL(rho_K^eta||pi_T), including the exact a(t) and b(t) coefficients.
2. appendix.tex:1558-1570 gives the pre-time-change differential inequality with the doubled residual term 2*sigma_eta^{-2}*dot t(s)^2 and the frozen-delta terms 2*Gamma(t(s))*eta^2*alpha'^{-1} and 2*Delta(t(s))*eta.
3. appendix.tex:1573-1583 defines K(t)=KL(hat rho_{s(t)}||pi_t)=KL(hat rho_{s(t)}||tilde pi_{s(t)}) and changes derivatives by dK/dt=dot{s}(t)*d/ds KL|_{s=s(t)}.
4. appendix.tex:1583 uses the constant inverse-schedule identity dot t(s(t))=dot{s}(t)^(-1) before rewriting the residual and frozen-delta coefficients.
5. appendix.tex:1586-1597 is the final t-time differential inequality; the lower proof must preserve the two residual coefficients and the dot{s}(t) multipliers on Gamma and Delta.
6. appendix.tex:1600 invokes lem:gronwall, whose endpoint laws, coefficient regularity, and exact display matching are not expanded in the source proof.

leanStepMap : List String (explicit)

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

Ordered data items

1. The theorem statement remains SALD.generalMovingTargetDiscreteStatementContract and SALD.generalVaSaldDiscreteContract; this packet changes neither.
2. SALD.generalMovingTargetDiscreteGronwallInstantiationContract records the source a(t), b(t), and Gronwall target before theorem-display side conditions.
3. SALD.generalMovingTargetDiscreteGronwallSideConditionContract records endpoint stitching, constant-schedule identities, coefficient regularity, residual/frozen coefficient audits, and display matching.
4. SALD.generalMovingTargetDiscreteConstantScheduleObligation supplies schedule and stitched-interval interfaces as obligations rather than new theorem assumptions.
5. Cycle-20 lower scalar helpers formalize only the real coefficient algebra after the inverse-schedule identity: SALD.generalMovingTargetDiscreteConstantScheduleSquareScalar, SALD.generalMovingTargetDiscreteResidualCoefficientRewriteScalar, SALD.generalMovingTargetDiscreteGammaCoefficientRewriteScalar, and SALD.generalMovingTargetDiscreteDeltaCoefficientRewriteScalar.
6. SALD.cycle20GeneralVaSaldDiscreteGronwallMiddleObligation keeps this middle source map synchronized with the lower target sald.general_moving_target_discrete.gronwall_side_conditions.
7. The cycle-17 and cycle-18 Gronwall scalar helpers are reusable only after theorem-specific interval-integrability and endpoint equalities are supplied; they do not prove this discrete general Gronwall step.

citedResultInterfaces : List String (explicit)

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

Ordered data items

1. lem:gronwall remains a local real-analysis obligation through sald.gronwall.integrating_factor and theorem-specific regularity/display side conditions.
2. lem:dv_variation remains source-cited only through the existing residual DV-energy obligations; this middle packet does not reopen or formalize DV.
3. eq:LSI-KL-FI remains the inherited density-test obligation used before the final Gronwall inequality.
4. No SLT theorem applies to endpoint stitching, constant-schedule coefficient rewrites, coefficient regularity, or exact display matching.

obligations : List String (explicit)

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

Ordered data items

1. sald.general_moving_target_discrete.cycle20_gronwall_middle
2. sald.general_moving_target_discrete.gronwall_side_conditions
3. sald.general_moving_target_discrete.gronwall_application
4. sald.general_moving_target_discrete.constant_schedule_stitching
5. sald.general_moving_target_discrete.kl_derivative
6. sald.general_moving_target_discrete.dv_m_energy
7. sald.general_moving_target_discrete.frozen_delta_cross_lip
8. sald.gronwall.integrating_factor

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. Preferred first sub-slice: constant-schedule coefficient rewrite from appendix.tex:1579-1597, especially dot{s}(t)*dot t(s(t))^2 = dot{s}(t)^(-1) and the unchanged Gamma/Delta multipliers.
3. Alternative one-slice targets are endpoint stitching for K(0)/K(T), coefficient regularity for a(t), b(t), or exact Gronwall-display matching against appendix.tex:1316-1347.
4. Keep the EM interpolation, frozen-delta lemma, residual DV finite-log-mgf witness, LSI bridge, KL derivative, and full Gronwall lemma as separate dependencies.
5. Do not add schedule, endpoint, regularity, or integrability assumptions to thm:general-moving-target-SALD-discrete; record blocked interfaces as obligations.

reviewerChecklist : List String (explicit)

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

Ordered data items

1. Every source step from appendix.tex:1573-1600 and the theorem display appendix.tex:1316-1347 is classified as Lean contract, cited result, or named obligation.
2. SALD.generalVaSaldDiscreteContract lists SALD.cycle20GeneralVaSaldDiscreteGronwallMiddleObligation and still keeps the theorem contractOnly.
3. SALD.generalVaSaldDiscreteProofDag contains ASTIS.SALD.general_moving_target_discrete.cycle20_middle_gronwall_bridge between the upper packet and gronwall_side_conditions blocks.
4. SALD.saldDependenciesForLabel "thm:general-moving-target-SALD-discrete" includes SALD.cycle20GeneralVaSaldMiddleContract and sald.general_moving_target_discrete.cycle20_gronwall_middle.
5. No theorem statement, source coefficient, source file selection, or analytic dependency status is changed.

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 cycle20GeneralVaSaldMiddleContract :
    GeneralVaSaldGuidedPathMiddleContract where
  guidedResidualSource := saldGuidedResidualSource
  generalContinuousSource := saldGeneralMovingTargetSource
  unifiedSource := saldUnifiedForwardKlSource
  discreteSource := saldGeneralMovingTargetDiscreteSource
  objective := "Translate the cycle-20 discrete general Gronwall/display bridge into a lower-ready map for sald.general_moving_target_discrete.gronwall_side_conditions, with endpoint stitching, constant-schedule coefficient rewrites, coefficient regularity, and theorem-display matching kept as explicit obligations."
  sourceStepMap := [
    "appendix.tex:1316-1347 fixes the theorem display for KL(rho_K^eta||pi_T), including the exact a(t) and b(t) coefficients.",
    "appendix.tex:1558-1570 gives the pre-time-change differential inequality with the doubled residual term 2*sigma_eta^{-2}*dot t(s)^2 and the frozen-delta terms 2*Gamma(t(s))*eta^2*alpha'^{-1} and 2*Delta(t(s))*eta.",
    "appendix.tex:1573-1583 defines K(t)=KL(hat rho_{s(t)}||pi_t)=KL(hat rho_{s(t)}||tilde pi_{s(t)}) and changes derivatives by dK/dt=dot{s}(t)*d/ds KL|_{s=s(t)}.",
    "appendix.tex:1583 uses the constant inverse-schedule identity dot t(s(t))=dot{s}(t)^(-1) before rewriting the residual and frozen-delta coefficients.",
    "appendix.tex:1586-1597 is the final t-time differential inequality; the lower proof must preserve the two residual coefficients and the dot{s}(t) multipliers on Gamma and Delta.",
    "appendix.tex:1600 invokes lem:gronwall, whose endpoint laws, coefficient regularity, and exact display matching are not expanded in the source proof."
  ]
  leanStepMap := [
    "The theorem statement remains SALD.generalMovingTargetDiscreteStatementContract and SALD.generalVaSaldDiscreteContract; this packet changes neither.",
    "SALD.generalMovingTargetDiscreteGronwallInstantiationContract records the source a(t), b(t), and Gronwall target before theorem-display side conditions.",
    "SALD.generalMovingTargetDiscreteGronwallSideConditionContract records endpoint stitching, constant-schedule identities, coefficient regularity, residual/frozen coefficient audits, and display matching.",
    "SALD.generalMovingTargetDiscreteConstantScheduleObligation supplies schedule and stitched-interval interfaces as obligations rather than new theorem assumptions.",
    "Cycle-20 lower scalar helpers formalize only the real coefficient algebra after the inverse-schedule identity: SALD.generalMovingTargetDiscreteConstantScheduleSquareScalar, SALD.generalMovingTargetDiscreteResidualCoefficientRewriteScalar, SALD.generalMovingTargetDiscreteGammaCoefficientRewriteScalar, and SALD.generalMovingTargetDiscreteDeltaCoefficientRewriteScalar.",
    "SALD.cycle20GeneralVaSaldDiscreteGronwallMiddleObligation keeps this middle source map synchronized with the lower target sald.general_moving_target_discrete.gronwall_side_conditions.",
    "The cycle-17 and cycle-18 Gronwall scalar helpers are reusable only after theorem-specific interval-integrability and endpoint equalities are supplied; they do not prove this discrete general Gronwall step."
  ]
  citedResultInterfaces := [
    "lem:gronwall remains a local real-analysis obligation through sald.gronwall.integrating_factor and theorem-specific regularity/display side conditions.",
    "lem:dv_variation remains source-cited only through the existing residual DV-energy obligations; this middle packet does not reopen or formalize DV.",
    "eq:LSI-KL-FI remains the inherited density-test obligation used before the final Gronwall inequality.",
    "No SLT theorem applies to endpoint stitching, constant-schedule coefficient rewrites, coefficient regularity, or exact display matching."
  ]
  obligations := [
    "sald.general_moving_target_discrete.cycle20_gronwall_middle",
    "sald.general_moving_target_discrete.gronwall_side_conditions",
    "sald.general_moving_target_discrete.gronwall_application",
    "sald.general_moving_target_discrete.constant_schedule_stitching",
    "sald.general_moving_target_discrete.kl_derivative",
    "sald.general_moving_target_discrete.dv_m_energy",
    "sald.general_moving_target_discrete.frozen_delta_cross_lip",
    "sald.gronwall.integrating_factor"
  ]
  lowerPacket := [
    "Target exactly SALD.generalMovingTargetDiscreteGronwallSideConditionContract / SALD.generalMovingTargetDiscreteGronwallSideConditionObligation / sald.general_moving_target_discrete.gronwall_side_conditions.",
    "Preferred first sub-slice: constant-schedule coefficient rewrite from appendix.tex:1579-1597, especially dot{s}(t)*dot t(s(t))^2 = dot{s}(t)^(-1) and the unchanged Gamma/Delta multipliers.",
    "Alternative one-slice targets are endpoint stitching for K(0)/K(T), coefficient regularity for a(t), b(t), or exact Gronwall-display matching against appendix.tex:1316-1347.",
    "Keep the EM interpolation, frozen-delta lemma, residual DV finite-log-mgf witness, LSI bridge, KL derivative, and full Gronwall lemma as separate dependencies.",
    "Do not add schedule, endpoint, regularity, or integrability assumptions to thm:general-moving-target-SALD-discrete; record blocked interfaces as obligations."
  ]
  reviewerChecklist := [
    "Every source step from appendix.tex:1573-1600 and the theorem display appendix.tex:1316-1347 is classified as Lean contract, cited result, or named obligation.",
    "SALD.generalVaSaldDiscreteContract lists SALD.cycle20GeneralVaSaldDiscreteGronwallMiddleObligation and still keeps the theorem contractOnly.",
    "SALD.generalVaSaldDiscreteProofDag contains ASTIS.SALD.general_moving_target_discrete.cycle20_middle_gronwall_bridge between the upper packet and gronwall_side_conditions blocks.",
    "SALD.saldDependenciesForLabel \"thm:general-moving-target-SALD-discrete\" includes SALD.cycle20GeneralVaSaldMiddleContract and sald.general_moving_target_discrete.cycle20_gronwall_middle.",
    "No theorem statement, source coefficient, source file selection, or analytic dependency status is changed."
  ]
  status := ProofStatus.obligation

/-- Cycle-24 upper packet for the continuous general VA-SALD Gronwall bridge.

This returns to `thm:general-moving-target-SALD` after the discrete and
forward-KL coefficient audits.  It selects only the endpoint/exponent
side-condition ledger behind the final Gronwall display, leaving the theorem
statement, unified specialization, discrete theorem, and analytic backends
unchanged.
-/