The kissing number in dimension five is at most 43

The kissing number τ5 is the largest number of non-overlapping unit balls in R^5 that can touch a common unit ball. The best bounds so far were 40 ≤ τ5 ≤ 44; the upper bound comes from a high-precision floating-point computation of the three-point bound by Mittelmann and Vallentin. We prove τ5 ≤ 43. The proof is an exact rational solution of the second level of the Lasserre hierarchy for spherical codes at degrees (d1, d2, δ) = (14, 20, 20). We use the formulation of de Laat, Leijenhorst and de Muinck Keizer and check the solution with their verification program. The certificate is obtained by rounding a convex combination of two dual iterates of the interior-point method, one on each side of the target value. Exact certificates for the same hierarchy also give A(4, arccos 1/3) ≤ 14, A(5, arccos 1/3) ≤ 22 and A(6, arccos 1/3) ≤ 34 for spherical codes, improving the previous bounds 15, 23 and 35; with a known code, A(4, arccos 1/3) = 14. The largest number of points at angle π/3 from a point of a kissing configuration in R^n equals A(n-1, arccos 1/3), so in R^5 this number is 14. We also prove the four bounds in the Lean proof assistant, assuming only the positivity of the zonal kernels given by the zonal polynomial files: once by checking the exact certificates with compiled code, and once, with every computation done by Lean's kernel, from the certificates rounded to 30 decimal places.