Fourth-moment conjectures for Rademacher sums

Let $\varepsilon_1,\ldots,\varepsilon_n$ be independent Rademacher signs, let $\sum a_i^2 = 1$ , write $S = \sum a_i\varepsilon_i$ and $q = \sum a_i^4$ , and let $\mu_p = \mathbb{E}|G|^p$ for a standard Gaussian $G$ . Two 2025 conjectures say that $q$ alone governs how far $S$ falls short of Gaussian.

Jakimiuk proved $\mathbb{E}|S|^p \le \mu_p - c_p q$ for $p \ge 3$ and conjectured the optimal constant is $c_p = \mu_p - 1$ throughout that range; separately he conjectured a dimension-free quadratic stability bound at the critical exponent $p = 3$ .

Baranski, Murawski, Nayar and Oleszkiewicz reduced the finite-dimensional $L_p/L_4$ Khintchine constant for $p \ge 5$ to $\sup_{x \ge 1}\|x + \varepsilon_1 + \cdots + \varepsilon_N\|_p / \|x + \varepsilon_1 + \cdots + \varepsilon_N\|_4$ and conjectured the supremum is attained at $x = 1$ - that is, the flat coefficient vector is the extremizer.

Result: Proved(see note)
Status: Resolved
AI contribution: AI co-developed
Method: Argument
Field: Khintchine inequalities
Posed by: Jacek Jakimiuk; Adam Barański, Daniel Murawski, Piotr Nayar, Krzysztof Oleszkiewicz
Year posed: 2025
Years open: 1y
Solved: 2026-08-18
Model: ChatGPT 5.6 Sol
Vendor: OpenAI
Collaborators: Peigan Gao, Jian Qian
Verification: Unreviewed
Publication: Preprint
Significance: 8 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Four results, and the first is partly a refutation. Jakimiuk conjectured $c_p = \mu_p - 1$ is optimal for every $p \ge 3$ ; the paper proves that for $p \ge 4$ and gives a counterexample for every $2 < p < 4$ , so the conjecture is false as posed and the corrected range is $p \ge 4$ . The witness is the two-coordinate vector $S_2 = (\varepsilon_1+\varepsilon_2)/\sqrt2$ .

The Baranski-Murawski-Nayar-Oleszkiewicz flat-point conjecture is proved outright, in the stronger form that $x \mapsto \|x+S_n\|_p/\|x+S_n\|_4$ is strictly decreasing on $[1,\infty)$ for every real $p \ge 5$ ; that range is the one they conjectured, so nothing is left over. Jakimiuk's second conjecture, dimension-free quadratic stability at $p = 3$ , is proved with an explicit constant, though the optimal constant there is only bracketed and stays open. The paper also records the exact fixed- $q$ moment and Laplace-transform envelopes, from which coefficient-sensitive tail bounds follow.

What the AI did

From the paper's "Statement of AI use": "Initial versions of the proofs of the sharp Gaussian stability inequality and the finite-dimensional $L_p$ - $L_4$ constant theorem were developed with assistance from ChatGPT 5.6 Sol. The authors checked and revised the arguments, take full responsibility for their mathematical content, and independently verified the extensions presented here." The abstract puts it as "The proofs are discovered with substantial assistance from ChatGPT 5.6 Sol."

Co-developed rather than discovered, and the disclosure is why: it credits the model with initial versions of two of the paper's four theorems, inside a paper the humans led, and this site gives a disclosure phrased as assistance the lower tier. The two theorems named are Theorem 1.1 and Theorem 1.3, which are the two conjectures being settled. Neither the counterexample below $p = 4$ nor the third-moment theorem is attributed to the model.

Verification

Unreviewed: a one-day-old arXiv preprint, unrefereed, with no independent endorsement, and none of the proofs were checked here.

The statements were, and all three survived. Every theorem here is an inequality between finite sums and one-dimensional integrals, so it can be evaluated directly and hunted for counterexamples. Theorem 1.1 was tested on 18,960 coefficient vectors - flat families, one spike plus a flat sea, and random vectors of every sparsity - across twelve exponents from 4 to 13, with no violation, and the stated equality case $q=1$ holds exactly. Proposition 4.3 requires the same inequality to fail for every $2 < p < 4$ : at $S_2 = (\varepsilon_1+\varepsilon_2)/\sqrt2$ it reduces to $2^{p/2} \le \mu_p + 1$ , which fails strictly across that range and is an exact equality at both endpoints. Theorem 1.3's strict monotonicity held in 32,000 evaluations over eight exponents and $n \le 10$ .

That is a check of the statements, not of the proofs, so the tier does not move. It is not nothing: a wrong constant or a misplaced threshold would have shown up at once. The script is scripts/check_rademacher.py in this site's repository.

Sources

Submitted by VibeGene on 19 Aug 2026

Fourth-moment conjectures for Rademacher sums

What was actually shown

What the AI did

Verification

Sources

Discussion