The Character Structure of the Collision Fluctuation
Universal across bases
The Mertens growth law holds in every base tested: 3, 6, 7, 10, 12. The fluctuation sum grows like -mu · log(log(x)) at s = 1 and diverges as a power law below s = 1 in every case. The critical boundary is s = 1 regardless of base. The gate width, the variance scaling, and the Mertens growth law are all base-independent.
Character decomposition
The fluctuation sum decomposes over Dirichlet characters mod b. In base 10, there are four characters. Each character component L_chi is an independent arithmetic object. The trivial character carries the bulk. The non-trivial characters carry smaller but nonzero corrections. The components are not proportional to classical L-values: they are new.
Spectral equidistribution
The eigenvalues of the cross-alignment matrix converge to a clean limit: lambda_k / Q approaches b^2 sin^2(pi k/b) / (pi k)^2. The deviation from this limit, summed over primes at s = 1, converges for every frequency k. And the limiting values are nearly the same across all frequencies: approximately 0.165 in base 10.
The fluctuation is spectrally flat. No harmonic mode concentrates the deviation. The crystal's departure from its asymptotic structure is uniformly distributed across frequencies.
Three orthogonal cuts
The collision fluctuation can be seen three ways: by spectral class (p mod b), by Dirichlet character, by eigenvalue frequency. The structure is consistent along each axis independently. This decomposition provides the framework for understanding why mu takes the value it does.
The paper
The Character Structure of the Collision Fluctuation (PDF)
.:.