Complete minimizer picture for Gamow's liquid drop model
Balls uniquely minimize the Gamow liquid drop energy at every volume up to a sharp threshold of about 3.51, and above that threshold no minimizer exists, closing a gap that partial results had narrowed from both sides without meeting.
- Lab
- Independent
- Model
- ChatGPT 5.6 Pro
- Field
- Mathematics
- Date
- 2026-08-12
- Human collaborators
- Otis Chodosh, Matilde Gianocca
Sources
Original work
Independent commentary
What was found
Gamow's liquid drop functional adds De Giorgi perimeter to a Coulomb repulsion term, and the central question in its literature is the fixed-volume minimization picture: for which volumes a minimizer exists, and whether it is the ball. This proves that balls uniquely minimize, up to translation and null sets, at every volume up to a threshold of 5 times 2 minus 2 to the two thirds, divided by 2 to the two thirds minus 1, roughly 3.51, and that above that threshold no minimizer exists at all. A corollary settles the minimal binding energy question of Frank and Lieb, identifying the infimum of energy per unit volume and the ball of volume 5/2 as the unique minimizer. The mechanism is a capacitary estimate sharpening an Agostiniani–Mazzieri monotonicity formula through Gauss–Bonnet, an improvement the authors note applies only to this particular weight and only in three dimensions.
Novelty check
The sharp threshold is the tracked open problem of the liquid drop literature, followed across calculus of variations, mathematical physics and geometric analysis, with a 2017 Notices of the AMS survey and partial results carrying Lieb, Otto, Figalli and Maggi. The prior state of the art is stated in the paper and checks out: minimality of the ball was known only up to volume 1 (Chodosh–Ruohoniemi, CPAM 2025) and nonexistence only from volume 7.5 upward (Schulz, arXiv:2608.09000, posted two days earlier), leaving the interval from 1 to 7.5 open. Frank and Nam had proved existence up to the threshold in 2021, and the new proof uses that result rather than superseding it. Uniqueness across the whole range and nonexistence immediately above the threshold are new.
Caveats and known objections
A preprint, not peer-reviewed, with no formalization. Verification is author-side: the authors state they checked and reworked the proof, which is not independent replication. Autonomy graded ai-led on the paper's own disclosure, which says the results were obtained by ChatGPT 5.6 Pro over a series of chats without significant assistance from the authors and that the fundamental strategy remains close to the model's original output; the authors checked the proof, reworked it and wrote the manuscript, and state the article contains no AI-written text.
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Entry history (1 event)
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Cite this entry
whataifound.org. (2026). Complete minimizer picture for Gamow's liquid drop model. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-08-12-liquid-drop-minimizersBibTeX
@misc{whataifound-independent-2026-minimizers,
title = {Complete minimizer picture for Gamow's liquid drop model},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Independent. Verification: Author verified. Autonomy: AI-led.},
url = {https://whataifound.org/finding/2026-08-12-liquid-drop-minimizers}
}