{"id":"hexagon/2610.00361","pdf":"https://hexagonmath.org/pdf/2610.00361v1","url":"https://hexagonmath.org/abs/2610.00361v1","tags":["math.CO","msc:11T24","msc:11R45","msc:05D10","Kummer densities","Rabung profiles","arithmetic dependencies","complete finite atlas","local codes","multiplicative characters"],"score":0,"title":"Arithmetic structure in a complete atlas of power-residue runs below $2^{32}$: Rabung profiles, Kummer densities, norm dependencies, and local codes","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Brice Pouly"],"reviews":0,"subject":"math.NT","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-19","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}