Golden Germ Window Zero Criterion

Abstract

RH confines golden-window zeros away from the residual zero set, with a conditional right-half-strip converse.

Theorem 1.1 (RH confines surviving window zeros to the pulled-back critical line).

$$RiemannHypothesis\Rightarrow \left(\forall G\in \mathbb{C}\to \mathbb{C},s\in \mathbb{C},\frac{1}{2\cdot {\phi }^3}<\Re s\Rightarrow \left(\Re s<\frac{1}{{\phi }^2}\Rightarrow \left(\mathrm{riemannZeta}\left({\phi }^2\cdot s\right)\cdot \mathrm{riemannZeta}\left({\phi }^3\cdot s\right)\cdot \mathrm{riemannZeta}{\left(2\cdot {\phi }^2\cdot s\right)}^{-1}\cdot \mathrm{riemannZeta}{\left(2\cdot {\phi }^3\cdot s\right)}^{-1}\cdot \mathrm{riemannZeta}\left(\left(2\cdot {\phi }^2+{\phi }^3\right)\cdot s\right)\cdot G\left(s\right)=0\Rightarrow \left(G\left(s\right)\ne 0\Rightarrow \Re s=\frac{1}{2\cdot {\phi }^2}\right)\right)\right)\right)$$

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

Source. Repository-derived.

Commentary.

The residual function G is arbitrary. Assuming the Riemann hypothesis, a zero of the displayed continued product in the open golden window lies on the pulled-back critical line whenever G is nonzero at that point.

The proof isolates the zeta factors. The phi-squared factor uses the frozen nontrivial-zero critical-line theorem; the remaining factors are excluded by the strict window bounds and Mathlib’s zeta nonvanishing theorem.

This conditional statement does not specialize G to the frozen third-order residual and does not establish the Riemann hypothesis.

Theorem 1.2 (Window confinement conditionally excludes right-half-strip zeros).

$$\forall G\in \mathbb{C}\to \mathbb{C},\left(\forall rho\in \mathbb{C},\mathrm{riemannZeta}\left(rho\right)=0\Rightarrow \left(\frac{1}{2}<\Re rho\Rightarrow \left(\Re rho<1\Rightarrow G\left(\frac{rho}{{\phi }^2}\right)\ne 0\right)\right)\right)\Rightarrow \left(\left(\forall s\in \mathbb{C},\frac{1}{2\cdot {\phi }^3}<\Re s\Rightarrow \left(\Re s<\frac{1}{{\phi }^2}\Rightarrow \left(\mathrm{riemannZeta}\left({\phi }^2\cdot s\right)\cdot \mathrm{riemannZeta}\left({\phi }^3\cdot s\right)\cdot \mathrm{riemannZeta}{\left(2\cdot {\phi }^2\cdot s\right)}^{-1}\cdot \mathrm{riemannZeta}{\left(2\cdot {\phi }^3\cdot s\right)}^{-1}\cdot \mathrm{riemannZeta}\left(\left(2\cdot {\phi }^2+{\phi }^3\right)\cdot s\right)\cdot G\left(s\right)=0\Rightarrow \left(G\left(s\right)\ne 0\Rightarrow \Re s=\frac{1}{2\cdot {\phi }^2}\right)\right)\right)\right)\Rightarrow \left(\forall rho\in \mathbb{C},\mathrm{riemannZeta}\left(rho\right)=0\Rightarrow \left(\frac{1}{2}<\Re rho\Rightarrow \left(\Re rho<1\Rightarrow False\right)\right)\right)\right)$$

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

Source. Repository-derived.

Commentary.

Here again G is arbitrary. The hResidual premise is an explicit unknown hypothesis: it requires G to survive at every pulled-back zeta zero in the right half of the critical strip.

Given that premise and the displayed window-confinement implication, scaling a hypothetical right-half-strip zero by one over phi squared produces a window zero. Confinement then forces the original real part to equal one half, a contradiction.

Because hResidual remains unknown, this theorem is only a conditional converse. It claims no progress toward proving the Riemann hypothesis.

References

• Truth anchor: D5/S3/Analytic/EulerGerm/GoldenGermWindowZeroCriterion.golden_window_zero_on_line_of_rh
• Truth anchor: D5/S3/Analytic/EulerGerm/GoldenGermWindowZeroCriterion.golden_window_zero_right_half_strip_converse