The Golden Phase Lock at Zeta Zeros
Abstract
Working at N = 120 = |2I| (the spinor double cover), we decompose the Hurwitz zeta function into Dirichlet character projections mod 5 at non-trivial zeros of ζ(s). The two order-4 characters χ₂ and χ₃ produce projections whose ratio is phase-locked to the line at angle arctan(1/φ) in the complex plane, where φ is the golden ratio. The phase takes exactly two values — arctan(1/φ) and π − arctan(1/φ) — flipping between them at successive zeros. Verified at 50 zeros to 40-digit precision with zero deviation. The identity arctan(1/φ) + arctan(φ) = π/2 connects the lock to the golden ratio's self-inverse property. The magnitude ratio |P(χ₂)|/|P(χ₃)| varies wildly (0.04 to 11.4), revealing the zeros as handoff points where dominance between golden conjugates changes. The phase lock is invisible at N = 60 (rotation group); the spinor double cover N = 120 is required to resolve the two golden characters. This is the critical-line signature that Paper 124 identified as living in the vertex fibre: arctan(1/φ) is the angle that the functional equation's Γ-factors impose at Re(s) = 1/2. Third in the three-paper RH sequence (123: sieve consumes φ+ψ, 124: bridge carries φ−ψ, 125: lock pins arctan(1/φ)). Eleventh in the S³/2I sequence.
Fields of Study & Subjects
Advanced Mathematical Theories and Applications
Analytic Number Theory Research
Advanced Mathematical Identities
Publication details
- Journal / source
- Zenodo (CERN European Organization for Nuclear Research)
- Publication date
- 14 March 2026
- Document type
- Preprint