The kissing number in dimension ten is at most 552

We prove that the kissing number in dimension ten is at most 552. The best upper bound we found is 553, from the semidefinite programming bound of Machado and de Oliveira Filho (Exp. Math. 2018); a computation of the same three-point bound at degree 20 by Leijenhorst and de Laat (Math. Program. Comput. 2024) gives about 553.055. Our proof is an exact rational certificate for the three-point bound of Bachoc and Vallentin at degree 21, with value 552.99004..., so the margin below 553 is about 0.010. The certificate is checked in full by a program with exact rational Bernstein coefficients on top boxes, rigorously bounded floating-point subdivision and exact Sturm counts; a separately written program in ball arithmetic checks the two solutions that are mixed to form it. The supplement contains the certificate, both verifiers with their logs, and a stand-alone check script that uses only the Python standard library. AI use: this note comes from an AI-assisted research lab directed by the author; the computations, the certificate, both verifiers and the text were produced with Claude (Anthropic) under the author's direction. The second verifier was written by a separate session of the same AI system, so this is not independent human review. Not yet peer reviewed. Contact: gameruchral71@gmail.com. Version 1.1 (13 Nov 2026) changes only two text files in the supplement; the paper, the certificate, the code and the logs are unchanged. In replay/CHECKER_VERDICT.md the smallest LDL pivot is 2.51e-22 (block G_7), not 1.2e-21, and the previous best bound 553 is credited to Machado and de Oliveira Filho, as in the paper. In the README, the stand-alone check takes about 1 s, not about 2 s. Version 1.2 (13 Nov 2026) corrects two sentences of the paper and the matching sentence of the abstract; the theorem, the certificate, the code and the logs are unchanged. Version 1 said that the first verifier "certifies the two solutions that are mixed"; neither is feasible on its own (the 2x2 block b of each is not positive semidefinite, and the degree-21 solution violates the three-point condition slightly); the first verifier proves the two conditions for them, and b is repaired only in the mixed certificate. The proof uses only the mixed certificate, which the second verifier checks in full. The stand-alone check takes about one second, not about two.