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The Sandier-Serfaty conjecture: the triangular lattice minimizes planar Coulomb renormalized energy

Sandier and Serfaty introduced a renormalized energy WW for infinite configurations of planar point charges against a uniform neutralizing background, defined through curl-free fields EE with div⁡E=2π(ν−m)\operatorname{div}E=2\pi(\nu-m), the logarithmic self-energy removed and energy taken per unit area. It governs microscopic vortex patterns in the Ginzburg-Landau model and limits of two-dimensional Coulomb gases. They proved that a minimum exists, that it is approximated by periodic fields, and that among Bravais lattices the triangular lattice is optimal (Theorems 1 and 2 of their 2012 paper). Their Conjecture 1 asks: is the triangular lattice the minimizer of WW over the full admissible class of fields, not only among lattices?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Coulomb gases; renormalized energy of Ginzburg-Landau vortex patterns
Posed by
Etienne Sandier and Sylvia Serfaty (Conjecture 1, Comm. Math. Phys. 313, 2012)
Year posed
2012
Years open
14y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
42 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every admissible curl-free field with unit background, W(E)≥W(E△)W(E)\ge W(E_\triangle), and inf⁡W=W(E△)\inf W=W(E_\triangle) is finite, with no separation, periodicity or Voronoi assumption on the competitor. Theorem 1.2 gives the finite version: on every square torus with n≥2n\ge2 points the renormalized energy is at least nW(E△)nW(E_\triangle). The proof builds one trial potential from Voronoi sectors on a minimizing torus configuration, with an exact identity leaving a nonnegative square, plus interval-arithmetic estimates. It does not classify minimizers (compact changes keep the energy per area) and does not treat the Riesz or higher-dimensional cases.

What the AI did

The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The proof combines a Voronoi-sector trial-potential argument with a finite interval-arithmetic certificate (C++ with Boost.Multiprecision) that the manuscript's README says to run; it uses no universal-optimality result as input. The companion 'Universal optimality of the triangular lattice' (September 23, 2026) reaches the logarithmic jellium minimum by a different route through the Petrache-Serfaty implication.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript was read against Sandier-Serfaty Conjecture 1. It states W(E)≥W(E△)W(E)\ge W(E_\triangle) for every cutoff family and every field in the admissible class A1\mathcal A_1 (density-one normalization, which the manuscript matches to the conjecture's background m=2πm=2\pi), with the triangular value finite and equal to the infimum. The proof was not refereed and the interval certificate was not re-run here. No Lean formalization in the release covers this manuscript (lean/docs/090.md lists only the universal-optimality and circle-packing statements). The manuscript itself notes the minimum statement does not classify all minimizing configurations.

Sources

Changelog1 change

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