Compact Hyperbolic Coxeter Six-dimensional Polytopes With Ten Facets

Abstract

We show that, up to isometry, there is exactly one compact hyperbolic Coxeter 6-polytope with 10 facets: the polytope $P_{6,10}$ whose Coxeter diagram appears as Figure 5 of Burcroff, attributed there to Bugaenko. Together with the results of Lannér, Kaplinskaja, Esselmann and Felikson-Tumarkin, and the independent classifications of Burcroff and Ma-Zheng in dimensions 4 and 5, this completes the classification of compact hyperbolic Coxeter $d$-polytopes with $d+4$ facets in every dimension. We enumerate all 387 candidate combinatorial types from the complete database of planar order types on 10 points via affine Gale duality, reduce to 11 types using two combinatorial consequences of Lannér's classification and the known classifications with $d+2$ facets and in dimension 5 with 9 facets, and decide those 11 by an exhaustive search over Coxeter labellings with no a priori bound on the dihedral angles. The search terminates with machine-checked exhaustion certificates, and the unique surviving Gram matrix is certified exactly over $\mathbb{Q}(\sqrt2,\sqrt5)$ and independently by CoxIter. The emptiness verdicts are exact as well: forward checking, by integer and real quadratic-field arithmetic, empties ten of the 11 types on its own, and every labelling that reaches a screen and is not accepted is refuted exactly over $\mathbb{Q}(\sqrt2,\sqrt3,\sqrt5)$, by the non-vanishing of a single determinant or, in 19 cases, by interval arithmetic on a compactified domain. No verdict rests on a floating-point tolerance or on a bound on the ultraparallel weights. The same code path, unchanged, reproduces the known censuses of 51 polytopes in dimension 5 and 348 in dimension 4. Code, data and certificates are publicly available. Most of the software was written by an AI assistant under the author's direction.