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

Corollary 1.3: $E_{\log}(n)=(\tfrac12-\log2)n^2-\tfrac n2\log n+C_{\mathrm{BHS}}n+o(n)$ 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, $d=2$ 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.

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. 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: The Brauchart-Hardin-Saff conjecture on the linear term of optimal logarithmic energy on the 2-sphere · Brauchart-Hardin-Saff log-energy conjecture

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

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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.

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

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Related in family: Universal optimality of the triangular lattice

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