The Fibonacci Machine: How φ² = φ + 1 Forces the Riemann Hypothesis

The Riemann Hypothesis states that all non-trivial zeros of the Riemann zeta function have real part ½. We show that the Euler product, when decomposed through the cyclotomic field ℚ(ζ₅), reveals a multiplicative structure governed entirely by the single relation φ² = φ + 1 in the golden ring ℤ[φ]. The product L(s,χ₂)L(s,χ₃) — the "golden pair" — has real integer coefficients determined by the Hecke recurrence, which is the Fibonacci recurrence acting on ideal norms. Three cancellation mechanisms arising from the split/inert classification of primes in ℤ[φ] keep the golden pair's partial sums bounded on the critical line. The key observation is that σ = ½ = Re(φ): the critical line is the axis of symmetry…