Golden Auxiliary Zeta Nonvanishing
Abstract
Riemann zeta is nonzero at the golden auxiliary point one over phi.
Theorem 1.1 (Riemann zeta is nonzero at the golden auxiliary point).
Proof. Machine-checked in Lean as D5/S3/Analytic/Isolation/GoldenAuxiliaryZetaNonzero.riemannZeta_golden_auxiliary_ne_zero (✓ std3). ∎
Source. Repository-derived.
Commentary.
This is the next pointwise step in the golden Euler germ extraction ladder of OACTC parts 580 and 581, on the RH-route O-5 control line. It closes the previously open special-value boundary at one over phi by proving that the zeta factor there cannot vanish.
The proof pairs adjacent terms of the Dirichlet eta series. Each real pair is strictly positive, while a derivative majorant gives absolute convergence for positive real part. An identity-theorem argument transports the usual zeta identity to the positive real axis. The frozen initial O-5 exponent power law identifies the golden coordinate used by the final specialization.
The exact bracket one half less than one over phi less than one records that the selected value is genuinely inside the critical strip. STOPPING JUSTIFICATION: this is one concrete nonvanishing value. It does not establish O-5, the Riemann hypothesis, any implication toward either claim, or a zero-free region around the point.
References
- Truth anchor:
D5/S3/Analytic/Isolation/GoldenAuxiliaryZetaNonzero.riemannZeta_golden_auxiliary_ne_zero