The Tammes problem for fifteen points

This is AI-generated research: the results, proofs and code were found and written by AI. Credit goes to all the humans whose work it builds on.

We prove that the largest possible minimal angular distance between fifteen points on the unit sphere is $\arccos u\approx 53.65785^\circ$, where $u$ is the root in $(1/2,7/10)$ of $13u^5-u^4+6u^3+2u^2-3u-1$. This value is attained by two configurations described by Buddenhagen and Kottwitz, whose contact graphs are not isomorphic, so the optimum is not unique. As Musin and Tarasov did for thirteen and fourteen points, a written reduction leads to plane graphs with at most fifteen vertices, and an interval computation rules out each of them except near the two known optima, where a local rigidity estimate concludes. The computation records its search trees as certificates, which are replayed. Lean 4 proves every step except the enumeration of the plane graphs and their refutation by the program, which enter the formal proof as hypotheses.

Corollary 1.2 of the paper answers in the negative, for $N=15$, Question 2 of Section 6 of Musin and Tarasov, Extremal problems of circle packings on a sphere and irreducible contact graphs, Proc. Steklov Inst. Math. 288 (2015), 117–131 (doi:10.1134/S0081543815010095; arXiv:1410.0744): is the contact graph of a maximal configuration of $N>5$ points unique up to isomorphism?

Lean 4 with Mathlib proves the main theorem and Corollary 1.2, with only the standard axioms, from three hypotheses: D2 for a set of plane graphs; for each of its graphs and each choice of hexagons, search trees for a choice in the same faces, on ranges of $d$ that cover the range of the proof, accepted by a family of interval procedures; and that these procedures are computed by the program in the sense of Definition B.1(5) of the paper. D2 holds for the list given by plantri and an output filter, and the trees are given by the program, whose certificates are replayed; the proof assumes that plantri, with the options of the paper, meets its description and that the filter keeps exactly the graphs of maximum degree at most 5, and that the program meets its model in the sense of Definition B.1(6) of the paper, which states what this asks of its arithmetic, its narrowings, its first level and its replay.

Code, Lean proofs and certificates: github.com/mt0-svg/tammes-15