Golden Germ Third-Order Factorization

Abstract

The third-order golden Euler factors give a unique continuation as a function above one over phi to the fifth power and retain the canonical germ on its original convergence half-plane.

Theorem 1.1 (Third-order factorization continues the golden germ past the phi-fifth line).

$$\begin{array}{r}\forall s\in \mathbb{C},p\in \mathrm{Primes}(\mathbb{N}),\\ x(p):=p^{-s\times {\phi }^2},y(p):=p^{-s\times {\phi }^3},\mathrm{Kp}(s,p):=(1-y(p{)}^2{)}^{-1}\times (1-x(p{)}^2\times y(p))\times (1-y(p))\times (1+x(p){)}^{-1}\times \sum _{v\in \mathbb{N}}{p}^{-s\times o5Beta(v)},\\ \forall s\in \mathbb{C},G3(s):=\prod _{p\in \mathrm{Primes}(\mathbb{N})}\mathrm{Kp}(s,p),\\ (\forall s\in \mathbb{C},\frac{1}{{\phi }^5}<\Re (s)\Rightarrow \mathrm{Summable}(p:\mathrm{Primes}(\mathbb{N})\mapsto |\mathrm{Kp}(s,p)-1|))\wedge \\ (\exists !Zphi:\{s\in \mathbb{C}\mid \frac{1}{{\phi }^5}<\Re (s)\}\to \mathbb{C},\\ (\forall s\in \mathbb{C},\frac{1}{{\phi }^2}<\Re (s)\Rightarrow \mathrm{Zphi}(s)=\prod _{p\in \mathrm{Primes}(\mathbb{N})}\sum _{v\in \mathbb{N}}{p}^{-s\times o5Beta(v)})\wedge (\forall s\in \mathbb{C},\frac{1}{{\phi }^5}<\Re (s)\Rightarrow \mathrm{Zphi}(s)=\mathrm{riemannZeta}({\phi }^2\times s)\times \mathrm{riemannZeta}({\phi }^3\times s)\times \mathrm{riemannZeta}(2\times {\phi }^2\times s{)}^{-1}\times \mathrm{riemannZeta}(2\times {\phi }^3\times s{)}^{-1}\times \mathrm{riemannZeta}((2\times {\phi }^2+{\phi }^3)\times s)\times G3(s))).\end{array}$$

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

Source. Repository-derived.

Commentary.

The frozen third-order ledger supplies the displayed local factor Kp and proves that its deviation from one is norm-summable when the real part exceeds one over phi to the fifth power. Consequently its prime product G3 is carried by a genuine Multipliable family.

On the original half-plane, the frozen second-order factorization is continued by extracting the Euler factors for twice phi-cubed and for two phi-squared plus phi-cubed. HasProd uniqueness identifies the resulting five-zeta expression with the canonical germ product. The displayed computation rule then determines the function uniquely throughout the larger half-plane.

This is the global third-order step in the golden Euler germ extraction staircase used in OACTC parts 580 and 581 and on the RH-route O-5 control line. It advances the previously open continuation boundary from one over phi to the fourth power to one over phi to the fifth power. The theorem asserts neither holomorphy nor meromorphic continuation on that region, and it does not assert nonvanishing, O-5, or the Riemann Hypothesis.

References

• Truth anchor: D5/S3/Analytic/EulerGerm/GoldenGermThirdOrderFactorization.golden_germ_third_order_factorization
• Dependency: D5/S3/Analytic/EulerGerm/GoldenGermSecondOrderFactorization
• Dependency: D5/S3/Analytic/EulerGerm/GoldenGermThirdOrderLedger