Purple Mathematics: A Visual Framework for Numerical Identity, Determination, and the On-Line Detection of Primes

Purple Mathematics is an expository, unifying framework that gives every positive integer a structured identity through one local operation, the *dig*: factor n² − 1 = (n − 1)(n + 1) and read its largest prime factor ρ(n), comparing it to a fixed wall W = 11. This sorts the integers into a Zero domain and an Infinity domain along a "bridge" between 0 and infinity, with the unit 1 as the fixed point of x ↦ 1/x. The Zero domain is finite (Størmer's theorem): exactly 49 integers, the largest being 19,601. The paper develops the resulting structure, the choice of the wall, a triangular wall whose heights are the prime gaps, an identity card separating identity fields from diagnostic measurements, two independent signed calibrations, a structural-stability interpretation layer, an exact on-line detection criterion, and one genuinely open analytic problem about an associated prime-driven walk. The framework is explicitly a lens over classical results (Størmer, Dickman, prime gaps, Pólya–Liouville oscillation, twin-smooth integers); no new theorem is claimed, and every component is mapped to its established counterpart.