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The Brauchart-Hardin-Saff conjecture on the linear term of optimal logarithmic energy on the 2-sphere

Let Elog⁡(n)E_{\log}(n) be the minimum over nn points on the unit sphere S2S^2 of −∑i≠jlog⁡∥yi−yj∥-\sum_{i\ne j}\log\|y_i-y_j\| (Smale's seventh problem concerns near-minimizers). The leading terms (12−log⁡2)n2−n2log⁡n(\tfrac12-\log2)n^2-\tfrac n2\log n are classical. Brauchart, Hardin and Saff (2012, Conjecture 4) conjectured that the next term is CBHSn+o(n)C_{\mathrm{BHS}}n+o(n) with the explicit constant CBHS=2log⁡2+12log⁡23+3log⁡(π/Γ(1/3))C_{\mathrm{BHS}}=2\log2+\tfrac12\log\tfrac23+3\log(\sqrt\pi/\Gamma(1/3)) coming from the triangular lattice. Betermin and Sandier (2018) proved that the linear coefficient exists and equals this value exactly when the triangular lattice minimizes the planar renormalized energy. Does Elog⁡(n)=(12−log⁡2)n2−n2log⁡n+CBHSn+o(n)E_{\log}(n)=(\tfrac12-\log2)n^2-\tfrac n2\log n+C_{\mathrm{BHS}}n+o(n) hold?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Discrete energy on the sphere; asymptotics of optimal logarithmic energy
Posed by
Johann S. Brauchart, Douglas P. Hardin and Edward B. Saff (Conjecture 4, 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
30 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Corollary 1.3: Elog⁡(n)=(12−log⁡2)n2−n2log⁡n+CBHSn+o(n)E_{\log}(n)=(\tfrac12-\log2)n^2-\tfrac n2\log n+C_{\mathrm{BHS}}n+o(n) with the Brauchart-Hardin-Saff constant, obtained from the triangular minimality of the planar Coulomb renormalized energy and the Betermin-Sandier equivalence. It is the logarithmic, d=2d=2 case only; the Riesz-energy cases of the Brauchart-Hardin-Saff conjectures and higher spheres are not addressed, and no explicit point configurations are produced.

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 result is Corollary 1.3 of the Coulomb renormalized-energy manuscript: it applies the Betermin-Sandier equivalence to that manuscript's main theorem after a normalization check.

Verification

No independent mathematician has checked this yet. Checked here: Corollary 1.3 of the principal manuscript was read against Conjecture 4 of Brauchart-Hardin-Saff in dimension two; it states the full asymptotic with the conjectured constant, derived from Theorem 1.1 (the Sandier-Serfaty conjecture) through Betermin-Sandier's Theorem 1.5. Its correctness therefore rests on Theorem 1.1, which is unreviewed. No Lean formalization covers it. The manuscript says this is an asymptotic value only, not a construction of near-optimal point sets or an algorithm for Smale's seventh problem.

Sources

Changelog1 change

Discussion