AutoSamplingTheory.SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract

Data definition / provenance and workflow record

Meaning and type

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

def generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract :
    GeneralMovingTargetDiscreteDvFiniteLogMgfWitnessContract

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.saldGeneralMovingTargetDiscreteResidualDvSource — audited data reference, not expanded and not a compiled dependency edge

theoremStatement : AutoSamplingTheory.SourceAnchor (explicit)

Nested provenance data.

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

dvMeasures : String (explicit)

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

appendix.tex:1544-1552 applies lem:dv_variation with nu=hat rho_s and mu=tilde pi_s=pi_{t(s)} under the general VA-SALD EM interpolation.

testFunction : String (explicit)

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

Z_s(x)=alpha*||m_{t(s)}(x)||^2, where m_t=v_t-c_t is the residual field from the continuous general theorem.

finiteAlpha0Assumption : String (explicit)

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

The discrete theorem imports the continuous general assumptions, including finite E_{alpha0}(pi_t,m_t), and allows alpha in (0,alpha0].

alphaMonotonicityBridge : String (explicit)

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

Reuse the continuous residual witness for finite log E_{pi_{t(s)}}[exp(alpha*||m_{t(s)}||^2)] before invoking DV on the EM interpolation law.

interpolationLawInterface : String (explicit)

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

hat rho_s is the law of the frozen EM interpolation and tilde pi_s=pi_{t(s)} on each stitched interval, as tracked by SALD.generalMovingTargetDiscreteDerivativeSideConditionContract.

commonSpaceAndAbsoluteContinuity : String (explicit)

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

The DV step requires hat rho_s and tilde pi_s on the same measurable space with the absolute-continuity/density interface needed for KL(hat rho_s||tilde pi_s).

measurabilityInterface : String (explicit)

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

m_{t(s)} and ||m_{t(s)}||^2 must be measurable under both hat rho_s and tilde pi_s; this remains part of the local analytic side-condition backend.

scalingStep : String (explicit)

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

After DV bounds E_{hat rho_s}[alpha*||m_{t(s)}||^2], divide by alpha>0 and rewrite the log-mgf as E_alpha(pi_{t(s)},m_{t(s)}).

outputBound : String (explicit)

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

||m_{t(s)}||_{L2(hat rho_s)}^2 <= alpha^(-1)*KL(hat rho_s||tilde pi_s)+E_alpha(pi_{t(s)},m_{t(s)}).

coefficientUse : String (explicit)

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

Multiplication by 2*sigma_eta^(-2)*dot t(s)^2 yields the doubled residual coefficients that become 2*sigma_eta(t)^(-2)*dot{s}(t)^(-1) after time change.

dependencies : List String (explicit)

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

Ordered data items

1. lem:dv_variation
2. def:alpha-complexity
3. SALD.saldDvFiniteLogMgfContract
4. sald.dv_variation.finite_log_mgf_interface
5. sald.general_moving_target.dv_finite_log_mgf_witness
6. sald.general_moving_target_discrete.em_interpolation_fp
7. sald.general_moving_target_discrete.derivative_side_conditions

sourceGaps : List String (explicit)

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

Ordered data items

1. the source applies DV in one display and does not isolate EM absolute-continuity or common-space facts
2. the continuous finite alpha0-complexity assumption for m_t is reused along t(s), but the alpha0-to-alpha monotonicity bridge is not stated separately
3. the proof must preserve the factor 2*sigma_eta^(-2)*dot t(s)^2 before the time-change rewrite

lowerPacket : List String (explicit)

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

Ordered data items

1. Target this contract or sald.general_moving_target_discrete.dv_finite_log_mgf_witness only.
2. Refine one backend: EM common-space/absolute-continuity, measurability of ||m_{t(s)}||^2, reuse of the continuous residual finite-log-mgf witness, positive-alpha scaling, or preservation of the doubled residual coefficient.
3. Do not add new hypotheses to thm:general-moving-target-SALD-discrete and do not modify the theorem display at appendix.tex:1316-1347.

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 generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract :
    GeneralMovingTargetDiscreteDvFiniteLogMgfWitnessContract where
  sourceBlock := saldGeneralMovingTargetDiscreteResidualDvSource
  theoremStatement := saldGeneralMovingTargetDiscreteSource
  dvMeasures := "appendix.tex:1544-1552 applies lem:dv_variation with nu=hat rho_s and mu=tilde pi_s=pi_{t(s)} under the general VA-SALD EM interpolation."
  testFunction := "Z_s(x)=alpha*||m_{t(s)}(x)||^2, where m_t=v_t-c_t is the residual field from the continuous general theorem."
  finiteAlpha0Assumption := "The discrete theorem imports the continuous general assumptions, including finite E_{alpha0}(pi_t,m_t), and allows alpha in (0,alpha0]."
  alphaMonotonicityBridge := "Reuse the continuous residual witness for finite log E_{pi_{t(s)}}[exp(alpha*||m_{t(s)}||^2)] before invoking DV on the EM interpolation law."
  interpolationLawInterface := "hat rho_s is the law of the frozen EM interpolation and tilde pi_s=pi_{t(s)} on each stitched interval, as tracked by SALD.generalMovingTargetDiscreteDerivativeSideConditionContract."
  commonSpaceAndAbsoluteContinuity := "The DV step requires hat rho_s and tilde pi_s on the same measurable space with the absolute-continuity/density interface needed for KL(hat rho_s||tilde pi_s)."
  measurabilityInterface := "m_{t(s)} and ||m_{t(s)}||^2 must be measurable under both hat rho_s and tilde pi_s; this remains part of the local analytic side-condition backend."
  scalingStep := "After DV bounds E_{hat rho_s}[alpha*||m_{t(s)}||^2], divide by alpha>0 and rewrite the log-mgf as E_alpha(pi_{t(s)},m_{t(s)})."
  outputBound := "||m_{t(s)}||_{L2(hat rho_s)}^2 <= alpha^(-1)*KL(hat rho_s||tilde pi_s)+E_alpha(pi_{t(s)},m_{t(s)})."
  coefficientUse := "Multiplication by 2*sigma_eta^(-2)*dot t(s)^2 yields the doubled residual coefficients that become 2*sigma_eta(t)^(-2)*dot{s}(t)^(-1) after time change."
  dependencies := [
    "lem:dv_variation",
    "def:alpha-complexity",
    "SALD.saldDvFiniteLogMgfContract",
    "sald.dv_variation.finite_log_mgf_interface",
    "sald.general_moving_target.dv_finite_log_mgf_witness",
    "sald.general_moving_target_discrete.em_interpolation_fp",
    "sald.general_moving_target_discrete.derivative_side_conditions"
  ]
  sourceGaps := [
    "the source applies DV in one display and does not isolate EM absolute-continuity or common-space facts",
    "the continuous finite alpha0-complexity assumption for m_t is reused along t(s), but the alpha0-to-alpha monotonicity bridge is not stated separately",
    "the proof must preserve the factor 2*sigma_eta^(-2)*dot t(s)^2 before the time-change rewrite"
  ]
  lowerPacket := [
    "Target this contract or sald.general_moving_target_discrete.dv_finite_log_mgf_witness only.",
    "Refine one backend: EM common-space/absolute-continuity, measurability of ||m_{t(s)}||^2, reuse of the continuous residual finite-log-mgf witness, positive-alpha scaling, or preservation of the doubled residual coefficient.",
    "Do not add new hypotheses to thm:general-moving-target-SALD-discrete and do not modify the theorem display at appendix.tex:1316-1347."
  ]
  status := ProofStatus.obligation