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Gamow liquid-drop minimizer conjecture

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.

Exact FrontierDelta

proved→proved

Scope and record

Occurred: Oct 29, 2026

Delta type: REGISTRY REVISION

Assumptions: VibeMathed verification: unreviewed. Publication: preprint. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Gamow liquid-drop minimizer conjecture · Liquid-drop minimizers

Confidence: Not scored

Registry verification: unreviewed · preprint · resolved

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Attribution

VibeMathed
registry · event recorded by

ChatGPT 5.6 Pro
model · ai model contributor · OpenAI

Otis Chodosh
human · human collaborator

Matilde Gianocca
human · human collaborator

Evidence vector

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Named evidence

VibeMathed record: Gamow liquid-drop minimizer conjecture

specialist registry · passed

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.

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Gamow liquid-drop minimizer conjecture

preprint · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Frank-Lieb (2015), source of the minimal binding energy question

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

Source ↗

Chodosh and Ruohoniemi, On minimizers in the liquid drop model (CPAM 2025) - previous best minimality range, V <= 1

paper · pending