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 to use 1/φ as a hash multiplier proven optimal via the Three-Distance Theorem.
The Fibonacci word eventually maximizes the LZ78 phrase count among all Sturmian sequences. The class of asymptotically co-maximal rotations is exactly the noble numbers.
The classical Lagrange spectrum read as a complete musical hierarchy of anti-consonance. φ sits dramatically isolated at the apex — 20.9% more anti-consonant than √2 (the tritone, rank 2).
Spherical polyhedra constructed from canonical embeddings of algebraic number fields. The icosahedron is the Galois solid of ℚ(φ) under A₅; the Alfredohedron is the Galois solid of ℚ(α) under A₄.