Arithmetic structure in a complete atlas of power-residue runs below $2^{32}$: Rabung profiles, Kummer densities, norm dependencies, and local codes
We study the arithmetic behind long runs in a complete census of power-residue colourings below $2^{32}$, covering every admissible order $2\le r\le9$. The dataset contains $203\,280\,221$ primes and $677\,591\,249$ profiles, with exact quadratic four-window counts at every odd prime. A bounded study of its deep tail identifies algebraic windows with fewer independent character conditions than their lengths suggest. We certify their power-class groups and conjugate-root overlaps, obtaining exact Kummer densities for fixed templates, including eight additional order-four models. In a…
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