Golden Germ Real-Axis Positivity

Abstract

The golden Euler germ prime product is a strictly positive real number throughout its full real convergence ray.

Theorem 1.1 (The golden germ product is positive on the convergence ray).

$$\forall sigma\in \mathbb{R},\frac{1}{{\phi }^2}<sigma\Rightarrow (\mathrm{Im}(\prod _{p\in \mathrm{Primes}(\mathbb{N})}\sum _{v\in \mathbb{N}}{p}^{-sigma\times o5Beta(v)})=0\wedge 0<\Re (\prod _{p\in \mathrm{Primes}(\mathbb{N})}\sum _{v\in \mathbb{N}}{p}^{-sigma\times o5Beta(v)})).$$

Proof. Machine-checked in Lean as D5/S3/Analytic/EulerGerm/GoldenGermRealAxisPositivity.golden_germ_real_axis_positivity (✓ std3). ∎

Source. Repository-derived.

Commentary.

This is the real-axis sign step in the golden Euler-germ extraction ladder of OACTC Parts 580 and 581, on the RH-route O-5 control line. It advances the previously unclosed boundary from positivity of the normalized factor to positivity of the original prime product on the entire real convergence ray.

For sigma greater than one over phi squared, the frozen multipliability theorem carries the prime product and the frozen factorization writes it as zeta at phi squared times sigma multiplied by the normalized factor. The zeta argument is greater than one, so both factors are positive real numbers.

The conclusion is confined to real sigma strictly inside the convergence ray. It does not assert positivity at the boundary, a complex zero-free region, the O-5 control statement, or the Riemann hypothesis.

References

• Truth anchor: D5/S3/Analytic/EulerGerm/GoldenGermRealAxisPositivity.golden_germ_real_axis_positivity
• Dependency: D5/S3/Analytic/EulerGerm/GoldenGermZetaFactorization