Kissing number in dimension 11: more than 604 spheres

Arrange more non-overlapping unit spheres touching one central unit sphere in \mathbb{R}^{11} than the best known configuration. Equivalently: find unit vectors x_1, \dots, x_m \in \mathbb{R}^{11} with

which is the statement that the centres are at least 60° apart.

Bar to beat. m \ge 605. The best known configuration has 604 spheres (improved in 2026); the best upper bound is 868, so the true value is unknown anywhere in [604, 868].

Submission format. m vectors of 11 exact rational coordinates.

Verifier sketch. Over the rationals the check is exact and scale-free: for every pair, 2\langle x_i, x_j\rangle \le \|x_i\| \|x_j\| compared by squaring, plus a sign check. That is \binom{605}{2} = 182{,}710 pairs, milliseconds.

Not written yet because of the tolerance question. Configurations in the literature are floating point, and a numerically "valid" arrangement can violate the constraint in the 15th digit. The verifier must either demand exact rationals (excluding most existing constructions) or certify a rounding with interval arithmetic. That choice is the whole design.

Known. Exact kissing numbers are known only in dimensions 1, 2, 3, 4, 8 and 24. Dimensions 11 and 12 both saw record configurations in 2026.

Why. Kissing configurations bound sphere packings, and dense packings in moderate dimensions are the geometry behind lattice codes and vector quantization. This listing is the prestige anchor of the set: real fame, no industrial revenue.

Ref: Wikipedia: Kissing number