All Primes Lead to 5 and 11: Dynamics of the Least Prime Quadratic Residue Map

For an odd prime p, let f(p) be the least odd prime q ≠ p modulo which p is a quadratic residue, equivalently the smallest q for which √p lies in the q-adic field Q_q. We show by an elementary descent argument that every orbit of f is absorbed by the 2-cycle {5, 11}, and that the first step is exactly geometric: a proportion 2^(-j) of primes lands on the j-th odd prime. As a consequence, the density of primes whose limit phase is 5 equals a constant c = 0.23037718…, whose binary digits are themselves the limit phases of the odd primes, so c encodes the fate of each prime while also measuring their average. The limit phase forgets the size of p and retains only h(c) ≈ 0.779 bits, mostly determined by p mod 15. Because the map sees only residues, random "fake primes" sieved by small primes reproduce all of its statistics, and it cannot distinguish a world of primes from a world of semiprimes: a small, explicit instance of Selberg's parity barrier.