The Tammes problem for sixteen points: a computer-assisted proof candidate
Year: 2026
Type: preprint
Abstract: Plain summary: the Tammes problem asks how to place N points on a sphere so that the closest pair is as far apart as possible. For N = 16 the best known arrangement (Sloane's D4d configuration) has, as far as we know, no published optimality proof. This record contains a computer-assisted proof candidate that this arrangement is optimal and unique, under stated hypotheses (four exact computations, and one further hypothesis for uniqueness), together with the code, the results and the exact commands to re-run every check. Independent review is welcome. We report a computer-assisted proof CANDIDATE that the largest possible minimal angular distance of 16 points on the unit sphere is d_16 = arccos u = 52.2443957527... degrees, where u is the root in (1/2, 7/10) of 23u^6 + 6u^5 + 5u^4 + 4u^3 - 3u^2 - 2u - 1, attained by exactly one configuration up to isometry (Sloane's D4d code, 32 contacts, no free point). The method is ported from the N = 15 package of mt0-svg (The Tammes problem for fifteen points, doi 10.5281/zenodo.23063365, Apache 2.0): contact-graph reduction, plantri enumeration, interval certificate trees, a Lean 4 reduction. The statement is conditional on four exact computations (D1 to D4, and one further hypothesis for uniqueness) that are hypotheses of the Lean theorems; the Lean implication is checked with Comparator. Machine checks (all pass in the final join): enumeration of all 4,256,414,982 class graphs (candidate contact graphs) at n = 16 under two splittings and an unrestricted plantri read; 3,528,656 first-level survivors (n = 14 to 16; 3,282,365 of them at n = 16), each refuted for every hexagon choice; every tree of the second level accepted by an independently written checker (3,593,867 trees, none rejected); a second, independently written first-level filter; deterministic reruns of the 18 parts where two earlier records differed; a third implementation of the free-point lemmas; an arbitrary-precision (MPFR) replay of a declared sample (181,326 graphs, 185,685 trees, all verified) and of every tree used by the tail (86 of 86 graphs). STATUS: this is a candidate. It has not been reviewed by any person outside the authors and nobody has accepted it. The certificate trees are not stored; they are regenerable by the published commands at the cost given in the paper (section 6). Please read review/READ-ME-FIRST.md in the archive first. The computation, code and text were produced by AI agents (Claude models of Anthropic) working under the direction of the author, who decided the goals and the publication policy. No human mathematician has reviewed the proof.
Author: Kępiński, Kamil
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221515238
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