Golden Germ Second Normalized Factor Regularity

Abstract

The second normalized golden germ product is holomorphic above one over phi to the fourth and is continuous and nonzero at the structural point one over phi cubed.

Theorem 1.1 (The second normalized factor is regular at the structural point one over phi cubed).

$$\begin{array}{r}H:\mathbb{C}\to \mathbb{C},\\ \forall s\in \mathbb{C},H(s):=\prod _{p\in \mathrm{Primes}(\mathbb{N})}(1-p^{-s\times {\phi }^3})\times (1+p^{-s\times {\phi }^2}{)}^{-1}\times \sum _{v\in \mathbb{N}}{p}^{-s\times o5Beta(v)},\\ \mathrm{AnalyticOnNhd}(\mathbb{C},H,\{s\in \mathbb{C}\mid \frac{1}{{\phi }^4}<\Re (s)\})\wedge \\ \mathrm{ContinuousAt}(H,\frac{1}{{\phi }^3})\wedge \\ H(\frac{1}{{\phi }^3})\ne 0.\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Analytic/Regularity/GoldenGermSecondNormalizedFactorRegularity.golden_germ_second_normalized_factor_regularity (✓ std3). ∎

Source. Repository-derived.

Commentary.

This theorem is the regularity step after the signed second-order factorization 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 open boundary by carrying the second normalized remainder across the structural point one over phi cubed.

For every real sigma above one over phi to the fourth, the proof splits each local series through its phi-fourth mode and builds a summable prime majorant for the normalized deviation. The locally uniform product theorem then gives holomorphy on the whole open half-plane.

At the structural point, each real germ-local series is a convergent sum of nonnegative terms with vacuum term one, hence is strictly positive. The two explicit real normalization factors are also nonzero. Frozen pointwise deviation summability then makes the infinite product nonzero.

The value one over phi cubed is the structural point given by D5.X_Frontier.Hearts.structuralPole. This theorem does not assert that the structural point is a pole; the pole conclusion is reserved for a later theorem. It does not claim regularity on the line with real part one over phi to the fourth, and does not prove or imply O-5 or the Riemann hypothesis.

References

• Truth anchor: D5/S3/Analytic/Regularity/GoldenGermSecondNormalizedFactorRegularity.golden_germ_second_normalized_factor_regularity
• Dependency: D5/S3/Analytic/EulerGerm/GoldenGermSecondOrderFactorization