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 quadratic family, eleven constant positions force twenty-six neutral positions at odd character orders. In two further families, norm identities generate all remaining higher-root relations. A length-\(53\) cubic realizes a three-wise independent binary code whose monochromatic probability is \(4096\) times that of independent signs; its local run law is computed exactly. A separate stratified million-profile ternary sample is analysed with the same finite catalogue filter applied to the data and to an independent-colour comparator. Finite census statements, selected explanations and asymptotic template laws concern distinct populations. No universal law of global maxima or finite-range error estimate for the template densities is asserted.