For irrationals α with constant continued fraction [0; a, a, a, …], the gap ratio of the Kronecker sequence {nα mod 1} takes exactly (a+1) distinct values forming an arithmetic sequence — a complete characterization of the noble numbers via the Three-Distance Theorem.
The sumset G + G of a geometric progression {1, r, r², …, r^(N−1)} has collisions if and only if r is a morphic number, identifying these ratios as the additive-minimal positive reals.
Knuth's recommendation of 1/φ as a hash multiplier, made exact through the Three-Distance Theorem: the largest gap between keys is always φ or φ² times the smallest, the tightest spread of any multiplier with a periodic continued fraction.
Second edition, September 2026: the proof rebuilt on the gap ratio, and the three missing figures added. The April edition remains available.
A Sturmian sequence of length L parses into at most 1.04·L^(2/3) LZ78 phrases, up to lower-order terms. The Fibonacci word fills 97 to 99 percent of that ceiling at every length tested, and the noble rotations run level with it.
Second edition, September 2026: the upper bound corrected, the lower bound left open, and the April maximality theorems withdrawn. The April edition remains available.
The classical Lagrange spectrum read as a musical hierarchy of anti-consonance. φ is first by the widest margin in the hierarchy, and √2, the tritone, is second at 20.9% below it.
Correction, September 2026: the abstract defines H by an infimum, which gives 1/φ² for φ. The value 1/√5 printed beside it belongs to the limit form used in Theorem 1.
Spherical polyhedra built from the canonical embeddings of algebraic number fields. The Alfredohedron, the Galois solid of ℚ(α) under A₄, is chiral, with twelve vertices and every distance in ℤ[α].
Second edition, September 2026: corrected distances and symmetry for the icosahedron and the Alfredohedron. The April edition remains available.