Golden Germ Third-Order Structural Zeros

Abstract

The two reciprocal zeta factors in the third-order golden germ create genuine simple structural zeros.

Theorem 1.1 (Both third-order denominator factors create simple zeros).

$$\begin{array}{r}\forall s\in \mathbb{C},G3(s):=\prod _{p\in \mathrm{Primes}(\mathbb{N})}(1-(p^{-s\times {\phi }^3}{)}^2{)}^{-1}\times (1-(p^{-s\times {\phi }^2}{)}^2\times p^{-s\times {\phi }^3})\times (1-p^{-s\times {\phi }^3})\times (1+p^{-s\times {\phi }^2}{)}^{-1}\times \sum _{v\in \mathbb{N}}{p}^{-s\times o5Beta(v)},\\ \forall s\in \mathbb{C},F3(s):=\mathrm{riemannZeta}({\phi }^2\times s)\times \mathrm{riemannZeta}({\phi }^3\times s)\times (\mathrm{riemannZeta}(2\times {\phi }^2\times s){)}^{-1}\times (\mathrm{riemannZeta}(2\times {\phi }^3\times s){)}^{-1}\times \mathrm{riemannZeta}((2\times {\phi }^2+{\phi }^3)\times s)\times G3(s),\\ z2:=\frac{1}{2\times {\phi }^2},z3:=\frac{1}{2\times {\phi }^3},\\ \mathrm{MeromorphicAt}(F3,z2)\wedge \mathrm{meromorphicOrderAt}(F3,z2)=1\wedge \mathrm{MeromorphicAt}(F3,z3)\wedge \mathrm{meromorphicOrderAt}(F3,z3)=1.\end{array}$$

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

Source. Repository-derived.

Commentary.

This theorem is the next local divisor step in the golden Euler germ extraction ladder of OACTC parts 580 and 581. It advances the third-order continuation by resolving the two structural points introduced by its reciprocal zeta factors.

At one over twice phi squared, the transported zeta arguments are one half, phi over two, phi, and one plus phi over two. At one over twice phi cubed, they are one over twice phi, one half, one over phi, and one over phi plus one half. The paired eta series supplies the positive-real nonvanishing facts below one.

The third normalized product is analytic and nonzero at both real points by the frozen regularity theorem. The golden auxiliary nonvanishing theorem handles the reciprocal factor at one over phi, while standard right-half-plane nonvanishing handles the transported arguments at least one.

The removable numerator riemannZeta1 rewrites each active reciprocal zeta factor as the first power of the local coordinate times an analytic nonzero multiplier. This proves meromorphy and exact order one, rather than relying on the totalized value at the pole.

STOPPING JUSTIFICATION: the conclusion concerns only these two local structural zeros of the displayed third-order continuation. It does not establish O-5, RH, a global zero classification, or an all-order extraction statement.

References

• Truth anchor: D5/S3/Analytic/Isolation/GoldenGermThirdOrderStructuralZeros.golden_germ_third_order_structural_zeros
• Dependency: D5/S3/Analytic/EulerGerm/GoldenGermThirdOrderFactorization
• Dependency: D5/S3/Analytic/Isolation/GoldenAuxiliaryZetaNonzero
• Dependency: D5/S3/Analytic/Isolation/GoldenGermStructuralSimplePole
• Dependency: D5/S3/Analytic/Regularity/GoldenGermThirdNormalizedFactorRegularity