Coincidence index and the support of rational orthogonal matrices

Abstract

For a rational orthogonal matrix, let Σ be the index of the integer vectors that it maps to integer vectors, and let m count its nonintegral columns. We prove the sharp universal inequality m ≤ 2Σ and classify its equality cases under independent signed row and column permutations. Two rigidity theorems describe the range near equality: above m = 3Σ/2, every nonintegral core is reflection-equivalent apart from four explicit exceptions; every core that is not reflection-equivalent satisfies the sharp additive bound m ≤ Σ + 4. The proofs combine coordinate root systems with energy estimates over the character group of the coincidence quotient. Applications determine the minimum index and all minimizers for even coincidence lattices, classify a sharp low-index window, and determine the second distinct index in sufficiently large dimension. A prime-level specialization proves the nonexistence assertion in a conjecture of Wang, Wang and Yu.