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For a measurable set $\Omega\subset\mathbb R^3$, let $$\mathcal E(\Omega)=P(\Omega)+\frac12\iint_{\Omega\times\Omega}\frac{dx\,dy}{|x-y|},$$ where $P$ is De Giorgi perimeter, and set $$V_*=5\frac{2-2^{2/3}}{2^{2/3}-1}\approx3.51.$$ The conjecture asks for the complete fixed-volume minimization picture. Chodosh and Gianocca prove that, for every $0<V\le V_*$, balls of volume $V$ uniquely minimize $\mathcal E$ among all measurable $\Omega$ with $|\Omega|=V$, up to translation and null sets; for $V>V_*$, no minimizer exists. Consequently, $$\inf_{0<|\Omega|<\infty}\frac{\mathcal E(\Omega)}{|\Omega|}=3\left(\frac{9\pi}{5}\right)^{1/3}=\frac92\left(\frac{8\pi}{15}\right)^{1/3},$$ with equality exactly for translates, modulo null sets, of the ball of volume $5/2$, equivalently radius $(15/(8\pi))^{1/3}$.
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The paper's AI-usage statement says that ChatGPT 5.6 Pro obtained the mathematical results over a series of chats without significant assistance from the authors. The fundamental proof strategy remained close to the model's output. Otis Chodosh and Matilde Gianocca then checked and reworked the proof and wrote the manuscript; they state that the article contains no AI-written text.
Provider: OpenAI · Prompt public: unknown · Independence: unknown
Checked here on 13 August 2026, the day after the preprint appeared. The AI-usage statement was confirmed verbatim in two places, the arXiv listing comment and the paper's own opening section. The LaTeX source was retrieved and the quantitative content rederived rather than trusted. The threshold came out independently from comparing one ball against two of half the volume as their separation grows, which favours splitting exactly when $V > 5(1-2^{1/3})/(2^{-2/3}-1)$ - the paper's $5(2-2^{2/3})/(2^{2/3}-1) \approx 3.5121$ after clearing radicals. The constant $|B_1|P(B_1)/D(B_1)=5$ checks against $D(B_R)=16\pi^2R^5/15$. The corollary does minimise at $V=5/2$ with value $3(9\pi/5)^{1/3}$, and the alternative form $\frac92(8\pi/15)^{1/3}$, which the submitter added and the paper does not state, is genuinely equal to it - both cube to $48.6\pi$. The identities the argument turns on expand as claimed, as do $2^{-2/3}(V_*+10)=V_*+5$ and the closing $1024<1296$ on $[6,8]$. Every cited source is real, with a resolving DOI. What was NOT checked is the capacitary estimate and the distributional Bochner lemma under it, which is where the new mathematics lives. The manuscript is one day old and unrefereed, and the authors checking their own reworked proof is not independent verification, so the tier stays Unreviewed.
Correctness: unknown · statement fidelity: unaudited · peer review: none
The complete fixed-volume picture, closing a gap that partial results had narrowed from both ends without meeting: balls uniquely minimize for every volume up to V_* = 3.51..., and above it no minimizer exists at all. Before this the best minimality range was V <= 1 (Chodosh-Ruohoniemi, 2025) and the best nonexistence bound V >= 7.5 (Schulz, posted two days earlier), so the open middle ran from 1 to 7.5. Frank-Nam had already proved existence up to V_*, and the new proof uses it; the fresh content is uniqueness of the ball across the whole range and nonexistence immediately above the threshold. A corollary settles the minimal binding energy question of Frank-Lieb: the infimum of E(Omega)/|Omega| is 3(9pi/5)^(1/3), attained exactly at balls of volume 5/2. The mechanism is a capacitary estimate that sharpens an Agostiniani-Mazzieri monotonicity formula using Gauss-Bonnet, an improvement the authors note applies only to this particular weight and only in three dimensions.
Source-reported tools: argument.
Independent: unknown · difference confidence: 0
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.