Arithmetic of Bohr Frequencies Governs Uniform Mixing in Randomly Timed Quantum Spin Chains

Randomizing the readout time averages the mode interference in a quantum walk. We ask when this makes the site populations of a uniformly coupled $XY$ chain exactly uniform for every initial site. For $N\ge 2$ spins, this is possible when $N+1$ is a power of two, a prime, or twice a prime. It is impossible when $15\mid (N+1)$, $21\mid (N+1)$, or $6\mid (N+1)$ with $N\ge 11$. Uniformity uniquely fixes the averaged cosine coherences according to the mirror parities of the modes. Equal Bohr frequencies can impose incompatible coherence values. We classify these parity collisions using vanishing sums of roots of unity and prove existence in the positive cases through a phase distribution on a torus.