Golden Germ Second-Order Factorization

Abstract

The golden germ has a canonical second-order continuation with two direct zeta factors, one reciprocal zeta factor, and an absolutely convergent tail.

Theorem 1.1 (The signed second-order factors continue the canonical golden germ).

$$\begin{array}{r}G3:\mathbb{C}\to \mathbb{C},\\ \forall s\in \mathbb{C},G3(s):=\prod _{p\in \mathrm{Primes}(\mathbb{N})}(1-p^{-s\times {\phi }^3})\times (1+p^{-s\times {\phi }^2}{)}^{-1}\times \sum _{v\in \mathbb{N}}{p}^{-s\times o5Beta(v)},\\ (\exists !Zqc2:\{s\in \mathbb{C}\mid \frac{1}{{\phi }^4}<\Re (s)\}\to \mathbb{C},\\ (\forall s\in \mathbb{C},\frac{1}{{\phi }^2}<\Re (s)\Rightarrow Zqc2(s)=\prod _{p\in \mathrm{Primes}(\mathbb{N})}\sum _{v\in \mathbb{N}}{p}^{-s\times o5Beta(v)})\wedge (\forall s\in \mathbb{C},\frac{1}{{\phi }^4}<\Re (s)\Rightarrow Zqc2(s)=\mathrm{riemannZeta}({\phi }^2\times s)\times \mathrm{riemannZeta}({\phi }^3\times s)\times \mathrm{riemannZeta}(2\times {\phi }^2\times s{)}^{-1}\times G3(s)))\wedge \\ (\forall s\in \mathbb{C},\frac{1}{{\phi }^4}<\Re (s)\Rightarrow \mathrm{Summable}(p:\mathrm{Primes}(\mathbb{N})\mapsto |(1-p^{-s\times {\phi }^3})\times (1+p^{-s\times {\phi }^2}{)}^{-1}\times \sum _{v\in \mathbb{N}}{p}^{-s\times o5Beta(v)}-1|)).\end{array}$$

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

Source. Repository-derived.

Commentary.

The continuation is uniquely determined by its displayed computation rule and agrees with the canonical prime product on the original absolute-convergence half-plane.

The normalized local factor cancels the phi-cubed mode and divides by one plus the phi-squared mode. Its deviation is absolutely summable above one over phi to the fourth power.

The reciprocal zeta factor is public in the formula. Fitted slopes, decimal thresholds, and finite-window error comparisons are empirical remarks outside the named theorem.

References

• Truth anchor: D5/S3/Analytic/EulerGerm/GoldenGermSecondOrderFactorization.golden_germ_second_order_factorization
• Dependency: D5/S3/Analytic/EulerGerm/GoldenGermZetaFactorization