The Golden Phase Lock at Zeta Zeros

University of Southampton

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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…

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Topics & keywords

Keywords
  • Golden ratio
  • Character (mathematics)
  • Phase (matter)
  • Zero (linguistics)
  • Vertex (graph theory)
  • Minimum phase
  • Spinor
  • Unit circle
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