Sharp Gaussian MGF deficits on a fixed symmetric support range

For a symmetric law with X²∈[3−√6,3+√6], put v=E[X²] and h=E[(X²−(3−√6))((3+√6)−X²)]. How large a uniform gap separates its moment generating function from exp(vt²/2) for all real t?

Why it matters and what counts as progress

Why it matters: A sharp deficit quantifies separation from the Gaussian moment-generating bound, beyond qualitative strict sub-Gaussianity.

What counts as progress: Prove an optimal quantitative lower bound for the deficit, including the h t⁴/24+t⁸/25200 target, and determine which coefficients can be simultaneously sharp.

How the problem developed

15 Oct 2026 · Sharp inequality [1][2][3]
JD Jones, with GPT-6 Pro and Codex assistance Established an optimal eighth-order gap between the Gaussian moment-generating function and that of a symmetric law with squared support in [3−√6,3+√6], including an optimal interiority coefficient.

AI contribution

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.

AI involvementAI-led

ImportanceNotable

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

Human understandingFew