ChatGPT-Assisted Work Settles Sharp Moment Conjectures for Rademacher Sums

In a manuscript dated August 18, 2026, Peigan Gao and Jian Qian develop sharp fourth-moment-sensitive inequalities for normalized Rademacher sums $S=\sum_i a_i\varepsilon_i$. Writing $q=\sum_i a_i^4$ and $\mu_p=\mathbb{E}|G|^p$ for a standard Gaussian $G$, they prove for every real $p\geq4$ that

$$\mathbb{E}|S|^p\leq \mu_p-(\mu_p-1)q,$$

with the optimal coefficient and a characterization of equality. This settles Jakimiuk's Gaussian-stability conjecture in its valid range; the paper notes that the originally proposed range fails for $2<p<4$.

The paper also identifies the exact fixed-$q$ envelope for $p\geq5$ through a one-spike-plus-Gaussian extremal law. In fixed dimension it proves, for $p\geq5$, that the sharp $L_p/L_4$ Khintchine ratio is attained uniquely by the flat coefficient vector, resolving a conjecture of Barański, Murawski, Nayar, and Oleszkiewicz. A separate theorem establishes Jakimiuk's dimension-free quadratic stability estimate at the critical exponent $p=3$.

Gao and Qian state that ChatGPT 5.6 Sol assisted with initial proofs of the sharp Gaussian-stability inequality and the finite-dimensional $L_p$–$L_4$ theorem. They report checking and revising those arguments, independently verifying the extensions, and taking responsibility for the mathematical content.