Gamow liquid-drop minimizer conjecture

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 $(15/(8\pi))^{1/3}$ .

Result: Proved(see note)
Status: Resolved
AI contribution: AI-discovered
Method: Argument
Field: Calculus of variations and geometric measure theory
Posed by: George Gamow (the functional, c. 1930); the sharp-threshold conjecture stated in the modern liquid-drop literature
Year posed: —
Years open: —
Solved: 2026-08-12
Model: ChatGPT 5.6 Pro
Vendor: OpenAI
Collaborators: Otis Chodosh, Matilde Gianocca
Verification: Unreviewed
Publication: Preprint
Significance: 35 / 100
Disclosed cost: —
Wikipedia: Not counted (article postdates the solution)

What was actually shown

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.

What the AI did

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.

Verification

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.

Sources

Submitted by October on 13 Aug 2026

Gamow liquid-drop minimizer conjecture

What was actually shown

What the AI did

Verification

Sources

Discussion