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AI-led proof gives an optimal eighth-order MGF gap

For every symmetric law with X² in [A,B], A=3−√6 and B=3+√6, set v=E[X²] and h=E[(X²−A)(B−X²)]. For every real t, exp(vt²/2)−E exp(tX)≥h t⁴/24+t⁸/25200. Both coefficients are optimal for this two-term bound. A positive coefficient on the intrinsic fourth deficit 3v²−E[X⁴] cannot be paired with the optimal eighth-order coefficient.

Sources for this record

What was already known

The published project record establishes the sharp support-ratio criterion and strict MGF domination, without this optimal quantitative gap. Earlier strict sub-Gaussian work gave necessary cumulant and sufficient moment conditions.

AI involvement AI-led

JD Jones directed the project. A GPT-6 Pro consultation supplied the sharp certificate; Codex agents developed and formalized the proof, with automated internal reviews.

The disclosure credits a GPT-6 Pro consultation with the sharp eighth-order certificate and Codex agents with proof integration and formalization under JD Jones’s direction.

Importance Notable

The result gives an optimal uniform quantitative separation from the Gaussian MGF bound in the normalized support bands.

Limitations

The law must be symmetric and have X² in the stated normalized bands. The sharp eighth-order term cannot be combined with a positive intrinsic-deficit coefficient. The reported verification run predates the cited documentation revision, which says formal sources were preserved; no human specialist review is documented.

Technical verification Formal (Lean 4/Mathlib, Comparator, NanoDa and con-ron with exact rational certificate checking)

The repository reports an exact-commit kernel and certificate run covering normalized_sharp_gap, both optimality declarations and the related strictness statement. The checked commit is 51c9ea6; the pinned documentation revision says the formal payload was preserved. This review did not compare the trees or rerun the suite.

Formal checking

Source-reported exact-commit statement comparison, kernel replay and certificate checks at 51c9ea6; not reproduced here.

Checker
Lean 4/Mathlib, Comparator, NanoDa, con-ron and project certificate checkers
Scope
normalized_sharp_gap, normalized_sharp_gap_strict, sharp_gap_coefficient_le, interiority_gap_coefficient_le and sharp_gap_intrinsic_coefficient_nonpos; repository states formal payload unchanged at the cited revision
Basis
Reported by the linked checker
Evidence source

Checking a formal statement does not establish that it matches the original problem.

Human understanding Few

The disclosure documents automated internal reviews but no named human explanation of the sharp certificate or tail argument. Few is the evidence default.

Correction and publication history

2026-10-18 · publication

The result gives an optimal uniform quantitative separation from the Gaussian MGF bound in the normalized support bands.

Read this published revision