AutoSamplingTheory.SALD.generalMovingTargetDvPositiveAlphaScalingContract

Data definition / provenance and workflow record

Meaning and type

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

def generalMovingTargetDvPositiveAlphaScalingContract :
    GeneralMovingTargetDvPositiveAlphaScalingContract

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

theoremStatement : AutoSamplingTheory.SourceAnchor (explicit)

Nested provenance data.

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

dvTestFunction : String (explicit)

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

Use the already-admissible DV test Z_t(x)=alpha*||m_t(x)||^2 from appendix.tex:885-893.

positivityInput : String (explicit)

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

The theorem range alpha in (0,alpha0] supplies alpha>0; this is the only scalar fact used to divide the DV inequality.

dvInequalityBeforeScaling : String (explicit)

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

After DV, alpha*E_{rho_{s(t)}}[||m_t||^2] <= KL(rho_{s(t)}||pi_t)+log E_{pi_t}[exp(alpha*||m_t||^2)].

divisionStep : String (explicit)

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

Divide the inequality by alpha>0 to preserve the inequality direction and get alpha^(-1)*K(t)+alpha^(-1)*log E_{pi_t}[exp(alpha*||m_t||^2)].

logMgfRewrite : String (explicit)

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

Rewrite alpha^(-1)*log E_{pi_t}[exp(alpha*||m_t||^2)] as mathfrak E_alpha(pi_t,m_t), exactly matching appendix.tex:891-895.

coefficientAudit : String (explicit)

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

Substitution into appendix.tex:899-907 must keep sigma_t^(-2)*dot{s}(t)^(-1)*alpha^(-1) on K(t) and sigma_t^(-2)*dot{s}(t)^(-1)*mathfrak E_alpha(pi_t,m_t) on the residual term.

unifiedSpecializationUse : String (explicit)

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

Under the unified specialization c_t<-u_t, the same scaling rewrite applies after m_t=w_t and preserves the main_body.tex:381 and 390 coefficients.

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.generalMovingTargetDvFiniteLogMgfWitnessContract
5. KLContract
6. Real positive scalar division/order algebra

sourceGaps : List String (explicit)

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

Ordered data items

1. appendix.tex:887-895 writes the divided inequality directly and does not isolate the alpha>0 order argument
2. the definition of mathfrak E_alpha must be unfolded with the same alpha and residual field m_t before the residual coefficient is multiplied in
3. main_body.tex:372-390 inherits this rewrite through m_t=w_t, but the source proof only says to specialize c_t<-u_t

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 generalMovingTargetDvPositiveAlphaScalingContract :
    GeneralMovingTargetDvPositiveAlphaScalingContract where
  sourceBlock := saldGeneralMovingTargetResidualDvSource
  theoremStatement := saldGeneralMovingTargetSource
  dvTestFunction := "Use the already-admissible DV test Z_t(x)=alpha*||m_t(x)||^2 from appendix.tex:885-893."
  positivityInput := "The theorem range alpha in (0,alpha0] supplies alpha>0; this is the only scalar fact used to divide the DV inequality."
  dvInequalityBeforeScaling := "After DV, alpha*E_{rho_{s(t)}}[||m_t||^2] <= KL(rho_{s(t)}||pi_t)+log E_{pi_t}[exp(alpha*||m_t||^2)]."
  divisionStep := "Divide the inequality by alpha>0 to preserve the inequality direction and get alpha^(-1)*K(t)+alpha^(-1)*log E_{pi_t}[exp(alpha*||m_t||^2)]."
  logMgfRewrite := "Rewrite alpha^(-1)*log E_{pi_t}[exp(alpha*||m_t||^2)] as mathfrak E_alpha(pi_t,m_t), exactly matching appendix.tex:891-895."
  coefficientAudit := "Substitution into appendix.tex:899-907 must keep sigma_t^(-2)*dot{s}(t)^(-1)*alpha^(-1) on K(t) and sigma_t^(-2)*dot{s}(t)^(-1)*mathfrak E_alpha(pi_t,m_t) on the residual term."
  unifiedSpecializationUse := "Under the unified specialization c_t<-u_t, the same scaling rewrite applies after m_t=w_t and preserves the main_body.tex:381 and 390 coefficients."
  dependencies := [
    "lem:dv_variation",
    "def:alpha-complexity",
    "SALD.saldDvFiniteLogMgfContract",
    "SALD.generalMovingTargetDvFiniteLogMgfWitnessContract",
    "KLContract",
    "Real positive scalar division/order algebra"
  ]
  sourceGaps := [
    "appendix.tex:887-895 writes the divided inequality directly and does not isolate the alpha>0 order argument",
    "the definition of mathfrak E_alpha must be unfolded with the same alpha and residual field m_t before the residual coefficient is multiplied in",
    "main_body.tex:372-390 inherits this rewrite through m_t=w_t, but the source proof only says to specialize c_t<-u_t"
  ]
  status := ProofStatus.obligation