The Golden Generator: Aperiodic Prime Generation from φ² = φ + 1

We demonstrate that the single algebraic relation φ² = φ + 1, encoded in the golden matrix [[1,1],[1,0]], generates all prime numbers and simultaneously classifies each prime by its decomposition in the golden ring ℤ[φ]. The Fibonacci test (golden matrix exponentiation mod p) detects primes via the divisibility F(p−1) ≡ 0 mod p for split primes and F(p+1) ≡ 0 mod p for inert primes. Residual pseudoprimes are eliminated by Miller-Rabin with witnesses 2 and 3 — the two smallest inert primes in ℤ[φ] — completing the golden toolkit. The method is verified to produce a perfect match against the Eratosthenes sieve through 1,000,000 (78,498 primes, zero errors); completeness for arbitrarily large n is conditional on the absence of Baillie-PSW counterexamples, verified to 2⁶⁴. The primes are not periodic (the quintic insolvability of A₅ forbids any periodic formula) and not random (the golden ratio generates them deterministically). We conjecture they constitute aperiodic order: a number-theoretic quasicrystal produced by the same golden ratio that generates Penrose tilings and physical icosahedrite. The same equation that generates the primes also governs the zeros of ζ(s) through the Fibonacci machine of Paper 126. An accompanying Python script (golden_generator.py) reproduces all results with no external dependencies.