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 certificates, and a second program written independently confirms it. 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), numbered as in its Russian version 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 two hypotheses checked by computation: D2, the completeness of the list of plane graphs, which plantri enumerates, and the search trees of the program, accepted by interval procedures that it computes and whose certificates the repository replays. Two steps are proved on paper: that the program computes these procedures, and D2 from the description of plantri. A second program, written separately, refutes every graph that the first level of the search leaves.

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