Golden Germ Second-Order Structural Residue

Abstract

The second-order golden germ has its explicit nonzero structural residue.

Theorem 1.1 (The structural residue is explicit and nonzero).

$$\begin{array}{r}H:\mathbb{C}\to \mathbb{C},\\ \forall s\in \mathbb{C},H(s):=\prod _{p\in \mathrm{Primes}(\mathbb{N})}(1-p^{-s\times {\phi }^3})\times (1+p^{-s\times {\phi }^2}{)}^{-1}\times \sum _{v\in \mathbb{N}}{p}^{-s\times o5Beta(v)},\\ F2:\mathbb{C}\to \mathbb{C},\\ \forall s\in \mathbb{C},F2(s):=\mathrm{riemannZeta}({\phi }^2\times s)\times \mathrm{riemannZeta}({\phi }^3\times s)\times (\mathrm{riemannZeta}(2\times {\phi }^2\times s){)}^{-1}\times H(s),\\ a:=\frac{1}{{\phi }^3},\\ \mathrm{MeromorphicAt}(F2,\frac{1}{{\phi }^3})\wedge \\ \mathrm{meromorphicOrderAt}(F2,\frac{1}{{\phi }^3})=-1\wedge \\ \mathrm{Tendsto}((s:\mathbb{C})\mapsto (s-\frac{1}{{\phi }^3})\times F2(s),\mathrm{nhdsWithin}(\frac{1}{{\phi }^3},\mathbb{C}\setminus \{\frac{1}{{\phi }^3}\}),\mathrm{nhds}(\mathrm{riemannZeta}(\frac{1}{\phi })\times (\mathrm{riemannZeta}(\frac{2}{\phi }){)}^{-1}\times H(\frac{1}{{\phi }^3})/{\phi }^3))\wedge \\ \mathrm{riemannZeta}(\frac{1}{\phi })\times (\mathrm{riemannZeta}(\frac{2}{\phi }){)}^{-1}\times H(\frac{1}{{\phi }^3})/{\phi }^3\ne 0.\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Analytic/Isolation/GoldenGermSecondOrderStructuralResidue.golden_germ_second_order_structural_residue (✓ std3). ∎

Source. Repository-derived.

Commentary.

This theorem is the residue step in the golden Euler germ extraction ladder of OACTC parts 580 and 581. The frozen structural simple-pole theorem fixes the point one over phi cubed and the exact meromorphic order, while this node closes the remaining local boundary by computing the coefficient.

The residue-one limit for Riemann zeta is transported through multiplication by phi cubed. The other factors are regular at the structural point: the squared zeta argument becomes one over phi, the doubled argument becomes two over phi, and the second normalized product H is continuous there.

GoldenAuxiliaryZetaNonzero supplies nonvanishing at one over phi. The standard right-half-plane theorem applies at two over phi because one is strictly less than two over phi, and GoldenGermSecondNormalizedFactorRegularity makes H nonzero. Together with the nonzero phi-cubed scale, these facts make the displayed residue nonzero.

STOPPING JUSTIFICATION: the conclusion concerns only the explicit second-order germ and its punctured neighborhood at one over phi cubed. It does not assert O-5, the Riemann hypothesis, any implication toward either statement, a zero-free region, or any all-orders extraction.

References

• Truth anchor: D5/S3/Analytic/Isolation/GoldenGermSecondOrderStructuralResidue.golden_germ_second_order_structural_residue
• Dependency: D5/S3/Analytic/EulerGerm/GoldenGermSecondOrderFactorization
• Dependency: D5/S3/Analytic/Isolation/GoldenAuxiliaryZetaNonzero
• Dependency: D5/S3/Analytic/Isolation/GoldenGermStructuralSimplePole
• Dependency: D5/S3/Analytic/Regularity/GoldenGermSecondNormalizedFactorRegularity