AutoSamplingTheory.SALD.cycle56DiscreteForwardKlGronwallLowerObligation

Data definition / provenance and workflow record

Meaning and type

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

def cycle56DiscreteForwardKlGronwallLowerObligation : ProofObligation

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.

id : String (explicit)

Stable obligation identifier.

sald.discrete_forward_kl.cycle56_gronwall_lower

statement : String (explicit)

Desired mathematical or workflow content as String; it is not a proposition in Prop and the def does not prove it.

Cycle 56 lower recovers the interrupted discrete forward-KL Gronwall route by compiling the scalar post-DV time-change handoff in appendix.tex:526-553. Once the EM/KL derivative backend, frozen-defect bound, LSI comparison, and DV velocity estimate supply the s-time inequality, SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar produces the exact t-time coefficient a(t)=dot{s}(t)*C_LSI(t)-dot{s}(t)^(-1)*alpha^(-1)-2*dot{s}(t)*eta^2*alpha'^(-1)*Gamma(t) and residual b(t)=dot{s}(t)^(-1)*E_alpha(pi_t,v_t)+2*dot{s}(t)*eta*Delta(t), while SALD.discreteForwardKlPointwiseGronwallInputOfPostDvTimeChanged packages that result pointwise in t for lem:gronwall.

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

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

status : AutoSamplingTheory.ProofStatus (explicit)

Stored ProofStatus, default obligation; even an explicitly stored formalized does not independently certify a Lean theorem.

AutoSamplingTheory.ProofStatus.obligation — stored label only; no proof certification

dependsOn : List String (explicit)

List of declared dependency names as strings; may mix theorem names, obligations, source labels, or descriptions. Not the compiled dependency DAG.

Ordered data items

1. SALD.cycle56DiscreteForwardKlSkeletonMiddleContract
2. SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation
3. SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar
4. SALD.discreteForwardKlPointwiseGronwallInputOfPostDvTimeChanged
5. SALD.discreteForwardKlTimeChangeSquareCoefficientRewriteScalar
6. SALD.discreteForwardKlGronwallInstantiationContract
7. SALD.discreteForwardKlGronwallAccumulationObligation
8. sald.discrete_forward_kl.gronwall_accumulation
9. SALD.cycle51DiscreteForwardKlDerivativeLowerObligation
10. sald.discrete_forward_kl.kl_derivative
11. SALD.discreteForwardKlDvFiniteLogMgfWitnessContract
12. sald.discrete_forward_kl.dv_velocity_bound
13. sald.forward_kl.schedule_time_change
14. lem:gronwall
15. lem:dv_variation
16. eq:LSI-KL-FI

note : String (explicit)

Recorded evidence/caveats; may distinguish a compiled scalar helper from still-open source analysis.

The compiled scalar theorem is real-order coefficient bookkeeping only. It does not prove EM conditional Fokker-Planck, KL differentiation, frozen-defect specialization, LSI/KL/FI, DV, Gronwall, stitched endpoint regularity, residual-exponent monotonicity, accumulated-error collection, or thm:forward-KL-discrete.

def cycle56DiscreteForwardKlGronwallLowerObligation : ProofObligation where
  id := "sald.discrete_forward_kl.cycle56_gronwall_lower"
  statement := "Cycle 56 lower recovers the interrupted discrete forward-KL Gronwall route by compiling the scalar post-DV time-change handoff in appendix.tex:526-553. Once the EM/KL derivative backend, frozen-defect bound, LSI comparison, and DV velocity estimate supply the s-time inequality, SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar produces the exact t-time coefficient a(t)=dot{s}(t)*C_LSI(t)-dot{s}(t)^(-1)*alpha^(-1)-2*dot{s}(t)*eta^2*alpha'^(-1)*Gamma(t) and residual b(t)=dot{s}(t)^(-1)*E_alpha(pi_t,v_t)+2*dot{s}(t)*eta*Delta(t), while SALD.discreteForwardKlPointwiseGronwallInputOfPostDvTimeChanged packages that result pointwise in t for lem:gronwall."
  source := saldForwardKlDiscreteGronwallSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.cycle56DiscreteForwardKlSkeletonMiddleContract",
    "SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation",
    "SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar",
    "SALD.discreteForwardKlPointwiseGronwallInputOfPostDvTimeChanged",
    "SALD.discreteForwardKlTimeChangeSquareCoefficientRewriteScalar",
    "SALD.discreteForwardKlGronwallInstantiationContract",
    "SALD.discreteForwardKlGronwallAccumulationObligation",
    "sald.discrete_forward_kl.gronwall_accumulation",
    "SALD.cycle51DiscreteForwardKlDerivativeLowerObligation",
    "sald.discrete_forward_kl.kl_derivative",
    "SALD.discreteForwardKlDvFiniteLogMgfWitnessContract",
    "sald.discrete_forward_kl.dv_velocity_bound",
    "sald.forward_kl.schedule_time_change",
    "lem:gronwall",
    "lem:dv_variation",
    "eq:LSI-KL-FI"
  ]
  note := "The compiled scalar theorem is real-order coefficient bookkeeping only. It does not prove EM conditional Fokker-Planck, KL differentiation, frozen-defect specialization, LSI/KL/FI, DV, Gronwall, stitched endpoint regularity, residual-exponent monotonicity, accumulated-error collection, or thm:forward-KL-discrete."

/-- Cycle-58 upper packet for the final unified/discrete general theorem
refresh.

This packet consumes the now-clean cycle-56 discrete forward-KL route and the
cycle-57 guided/general route.  It keeps the cycle focus on theorem-level
closure for `thm:unified-forward-KL` and
`thm:general-moving-target-SALD-discrete`; broad cited-theory backfill remains
deferred until the final theorem skeleton route is accepted.
-/