AutoSamplingTheory.SALD.forwardKlDependencyChainAuditContract

Data definition / provenance and workflow record

Meaning and type

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

def forwardKlDependencyChainAuditContract :
    ForwardKlDependencyChainAuditContract

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

postYoungInequality : String (explicit)

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

appendix.tex:210-217 combines the KL derivative terms after Young: d/ds KL(rho_s||tilde_pi_s) <= -(1/2)*FI(rho_s||tilde_pi_s)+(1/2)*||tilde_v_s||_{L2(rho_s)}^2.

lsiCoefficientStep : String (explicit)

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

Using eq:LSI-KL-FI, (1/2)*FI(rho_s||tilde_pi_s) >= C_LSI(t(s))*KL(rho_s||tilde_pi_s), giving the source coefficient -C_LSI(t(s))*K_s.

timeChangeStep : String (explicit)

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

appendix.tex:218-228 multiplies by dot{s}(t) and uses dot{s}(t)*dot{t}(s(t))^2=dot{s}(t)^(-1), producing the velocity term (1/2)*dot{s}(t)^(-1)*||v_t||_{L2(rho_{s(t)})}^2.

dvInstantiation : String (explicit)

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

appendix.tex:230-241 applies lem:dv_variation with nu=rho_{s(t)}, mu=pi_t, and Z=alpha*||v_t||^2, rewriting alpha^(-1)*log E_{pi_t}[exp(alpha||v_t||^2)] as E_alpha(pi_t,v_t).

scalarDifferentialInequality : String (explicit)

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

dK/dt <= -(dot{s}(t)*C_LSI(t)-(1/2)*dot{s}(t)^(-1)*alpha^(-1))*K(t)+(1/2)*dot{s}(t)^(-1)*E_alpha(pi_t,v_t).

gronwallA : String (explicit)

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

a(t)=dot{s}(t)*C_LSI(t)-(1/2)*dot{s}(t)^(-1)*alpha^(-1).

gronwallB : String (explicit)

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

b(t)=(1/2)*dot{s}(t)^(-1)*E_alpha(pi_t,v_t).

exponentSplit : String (explicit)

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

appendix.tex:244-250 splits exp(-int a) into exp(-int dot{s}*C_LSI)*exp(int (1/2)*dot{s}^(-1)*alpha^(-1)).

residualExponentSimplification : String (explicit)

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

appendix.tex:248-251 drops the nonpositive contribution -int_t^T dot{s}(u)*C_LSI(u) du in the residual exponential, using C_LSI(u)>=0 and dot{s}(u)>0.

terminalEndpointBridge : String (explicit)

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

The theorem bound identifies K(T)=KL(rho_S||pi_T) and K(0)=KL(rho_0||pi_0), requiring S=s(T), s(0)=0, and tilde_pi_{s(t)}=pi_t.

sourceLineLedger : List String (explicit)

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

Ordered data items

1. appendix.tex:210-217: the post-Young inequality leaves exactly -(1/2)*FI plus (1/2)*||tilde v_s||^2, then eq:LSI-KL-FI turns the FI term into -C_LSI(t(s))*K_s.
2. appendix.tex:218-228: the time change multiplies by dot{s}(t) and rewrites dot{s}(t)*dot t(s(t))^2 as dot{s}(t)^(-1), preserving the factor 1/2.
3. appendix.tex:230-241: DV with Z=alpha*||v_t||^2 contributes (1/2)*dot{s}(t)^(-1)*alpha^(-1)*K(t) and (1/2)*dot{s}(t)^(-1)*E_alpha(pi_t,v_t).
4. appendix.tex:244-247: Gronwall is applied with a(t)=dot{s}(t)*C_LSI(t)-(1/2)*dot{s}(t)^(-1)*alpha^(-1) and b(t)=(1/2)*dot{s}(t)^(-1)*E_alpha(pi_t,v_t).
5. appendix.tex:248-252 and main_body.tex:243-246: the initial exponent is split into the LSI contraction factor and the positive alpha factor, while the residual exponent drops only the nonpositive LSI contribution.

scalarSideConditions : List String (explicit)

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

Ordered data items

1. alpha>0 and alpha<=alpha0 are required for alpha^(-1) and for the finite log-mgf witness from the source alpha0-complexity assumption.
2. dot{s}(t)>0 and dot{t}(s(t))=dot{s}(t)^(-1) are required by the inverse-schedule time-change and coefficient rewrite.
3. C_LSI(t)>=0 is used only in the residual-exponent drop, not to change the differential inequality or theorem statement.
4. K(t), a(t), and b(t) need the differentiability/continuity or interval-integrability required by lem:gronwall.
5. Endpoint identities s(0)=0, S=s(T), and tilde_pi_{s(t)}=pi_t are required before rewriting K(0) and K(T).
6. rho_s, tilde_pi_s, and pi_t must live on a common state space with enough density and boundary regularity for the KL derivative and DV instantiation.

sourceDependencyClassification : List String (explicit)

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

Ordered data items

1. eq:LSI-KL-FI: local density-test obligation using phi=sqrt(rho/pi), not a theorem-level assumption.
2. lem:dv_variation: external-cited result with local finite-log-mgf/common-space instantiation; SLT entropy-duality remains reference-only.
3. lem:gronwall: local real-analysis obligation over interval integrals and endpoint-safe derivative semantics.
4. KL derivative and Fokker--Planck identity: local lemma plus source-contract gap for density, boundary, and differentiation-under-integral assumptions.
5. Final exponent split and endpoint bridge: local scalar/integral algebra, tracked by the gronwall side-condition obligation.

dependencies : List String (explicit)

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

Ordered data items

1. eq:LSI-KL-FI
2. def:alpha-complexity
3. lem:dv_variation
4. lem:gronwall
5. sald.forward_kl.moving_target_dependency_chain
6. sald.forward_kl.kl_derivative
7. sald.forward_kl.dv_energy_bound
8. sald.forward_kl.gronwall_application

sourceGaps : List String (explicit)

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

Ordered data items

1. the paper does not isolate the nonnegativity/positivity facts used in the residual exponent simplification
2. the theorem statement does not state the endpoint identities S=s(T) and s(0)=0 as standalone hypotheses
3. the finite-log-mgf monotonicity from alpha <= alpha0 to the DV test function is used implicitly

lowerPacket : List String (explicit)

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

Ordered data items

1. First formalize the scalar coefficient algebra from the derivative and DV obligations before attempting a theorem-level proof.
2. Keep the Young coefficient 1/2, the LSI coefficient C_LSI, and the DV contribution (1/2)*dot{s}^(-1)*alpha^(-1) unchanged.
3. Do not replace the final residual exponent simplification with a different Gronwall normalization.

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 forwardKlDependencyChainAuditContract :
    ForwardKlDependencyChainAuditContract where
  sourceBlock := saldForwardKlDependencyChainSource
  postYoungInequality := "appendix.tex:210-217 combines the KL derivative terms after Young: d/ds KL(rho_s||tilde_pi_s) <= -(1/2)*FI(rho_s||tilde_pi_s)+(1/2)*||tilde_v_s||_{L2(rho_s)}^2."
  lsiCoefficientStep := "Using eq:LSI-KL-FI, (1/2)*FI(rho_s||tilde_pi_s) >= C_LSI(t(s))*KL(rho_s||tilde_pi_s), giving the source coefficient -C_LSI(t(s))*K_s."
  timeChangeStep := "appendix.tex:218-228 multiplies by dot{s}(t) and uses dot{s}(t)*dot{t}(s(t))^2=dot{s}(t)^(-1), producing the velocity term (1/2)*dot{s}(t)^(-1)*||v_t||_{L2(rho_{s(t)})}^2."
  dvInstantiation := "appendix.tex:230-241 applies lem:dv_variation with nu=rho_{s(t)}, mu=pi_t, and Z=alpha*||v_t||^2, rewriting alpha^(-1)*log E_{pi_t}[exp(alpha||v_t||^2)] as E_alpha(pi_t,v_t)."
  scalarDifferentialInequality := "dK/dt <= -(dot{s}(t)*C_LSI(t)-(1/2)*dot{s}(t)^(-1)*alpha^(-1))*K(t)+(1/2)*dot{s}(t)^(-1)*E_alpha(pi_t,v_t)."
  gronwallA := "a(t)=dot{s}(t)*C_LSI(t)-(1/2)*dot{s}(t)^(-1)*alpha^(-1)."
  gronwallB := "b(t)=(1/2)*dot{s}(t)^(-1)*E_alpha(pi_t,v_t)."
  exponentSplit := "appendix.tex:244-250 splits exp(-int a) into exp(-int dot{s}*C_LSI)*exp(int (1/2)*dot{s}^(-1)*alpha^(-1))."
  residualExponentSimplification := "appendix.tex:248-251 drops the nonpositive contribution -int_t^T dot{s}(u)*C_LSI(u) du in the residual exponential, using C_LSI(u)>=0 and dot{s}(u)>0."
  terminalEndpointBridge := "The theorem bound identifies K(T)=KL(rho_S||pi_T) and K(0)=KL(rho_0||pi_0), requiring S=s(T), s(0)=0, and tilde_pi_{s(t)}=pi_t."
  sourceLineLedger := [
    "appendix.tex:210-217: the post-Young inequality leaves exactly -(1/2)*FI plus (1/2)*||tilde v_s||^2, then eq:LSI-KL-FI turns the FI term into -C_LSI(t(s))*K_s.",
    "appendix.tex:218-228: the time change multiplies by dot{s}(t) and rewrites dot{s}(t)*dot t(s(t))^2 as dot{s}(t)^(-1), preserving the factor 1/2.",
    "appendix.tex:230-241: DV with Z=alpha*||v_t||^2 contributes (1/2)*dot{s}(t)^(-1)*alpha^(-1)*K(t) and (1/2)*dot{s}(t)^(-1)*E_alpha(pi_t,v_t).",
    "appendix.tex:244-247: Gronwall is applied with a(t)=dot{s}(t)*C_LSI(t)-(1/2)*dot{s}(t)^(-1)*alpha^(-1) and b(t)=(1/2)*dot{s}(t)^(-1)*E_alpha(pi_t,v_t).",
    "appendix.tex:248-252 and main_body.tex:243-246: the initial exponent is split into the LSI contraction factor and the positive alpha factor, while the residual exponent drops only the nonpositive LSI contribution."
  ]
  scalarSideConditions := [
    "alpha>0 and alpha<=alpha0 are required for alpha^(-1) and for the finite log-mgf witness from the source alpha0-complexity assumption.",
    "dot{s}(t)>0 and dot{t}(s(t))=dot{s}(t)^(-1) are required by the inverse-schedule time-change and coefficient rewrite.",
    "C_LSI(t)>=0 is used only in the residual-exponent drop, not to change the differential inequality or theorem statement.",
    "K(t), a(t), and b(t) need the differentiability/continuity or interval-integrability required by lem:gronwall.",
    "Endpoint identities s(0)=0, S=s(T), and tilde_pi_{s(t)}=pi_t are required before rewriting K(0) and K(T).",
    "rho_s, tilde_pi_s, and pi_t must live on a common state space with enough density and boundary regularity for the KL derivative and DV instantiation."
  ]
  sourceDependencyClassification := [
    "eq:LSI-KL-FI: local density-test obligation using phi=sqrt(rho/pi), not a theorem-level assumption.",
    "lem:dv_variation: external-cited result with local finite-log-mgf/common-space instantiation; SLT entropy-duality remains reference-only.",
    "lem:gronwall: local real-analysis obligation over interval integrals and endpoint-safe derivative semantics.",
    "KL derivative and Fokker--Planck identity: local lemma plus source-contract gap for density, boundary, and differentiation-under-integral assumptions.",
    "Final exponent split and endpoint bridge: local scalar/integral algebra, tracked by the gronwall side-condition obligation."
  ]
  dependencies := [
    "eq:LSI-KL-FI",
    "def:alpha-complexity",
    "lem:dv_variation",
    "lem:gronwall",
    "sald.forward_kl.moving_target_dependency_chain",
    "sald.forward_kl.kl_derivative",
    "sald.forward_kl.dv_energy_bound",
    "sald.forward_kl.gronwall_application"
  ]
  sourceGaps := [
    "the paper does not isolate the nonnegativity/positivity facts used in the residual exponent simplification",
    "the theorem statement does not state the endpoint identities S=s(T) and s(0)=0 as standalone hypotheses",
    "the finite-log-mgf monotonicity from alpha <= alpha0 to the DV test function is used implicitly"
  ]
  lowerPacket := [
    "First formalize the scalar coefficient algebra from the derivative and DV obligations before attempting a theorem-level proof.",
    "Keep the Young coefficient 1/2, the LSI coefficient C_LSI, and the DV contribution (1/2)*dot{s}^(-1)*alpha^(-1) unchanged.",
    "Do not replace the final residual exponent simplification with a different Gronwall normalization."
  ]
  status := ProofStatus.obligation