Golden Germ Nonvanishing Above Three Fifths

Abstract

The prime-2 golden local factor and the full golden Euler product are nonzero when the real part is at least three fifths.

Theorem 1.1 (The prime-2 local factor is nonzero above three fifths).

$$\forall s\in \mathbb{C},\frac{3}{5}\le \Re (s)\Rightarrow \sum _{v\in \mathbb{N}}{2}^{-s\times o5Beta(v)}\ne 0.$$

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

Source. Repository-derived.

Commentary.

This is the prime-2 step in the golden Euler-germ extraction ladder of OACTC Parts 580 and 581, on the RH-route O-5 control line. It advances the explicit prime-2 nonvanishing boundary from real part two thirds to real part three fifths.

At the new endpoint, exact rational power bounds give the first tail coefficient below seventeen fiftieths and the geometric ratio below thirty-three fiftieths. Their sum is strictly below one, so the excited tail cannot cancel the vacuum term.

The remaining strip between one over phi squared and three fifths is not decided. The statement neither asserts a local zero below three fifths nor proves O-5 or the Riemann hypothesis.

Theorem 1.2 (The full golden Euler product is nonzero above three fifths).

$$\forall s\in \mathbb{C},\frac{3}{5}\le \Re (s)\Rightarrow \prod _{p\in \mathrm{Primes}(\mathbb{N})}\sum _{v\in \mathbb{N}}{p}^{-s\times o5Beta(v)}\ne 0.$$

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

Source. Repository-derived.

Commentary.

Odd-prime local factors were already nonzero throughout the open convergence half-plane. Combining that frozen result with the improved prime-2 estimate makes every local factor nonzero when the real part is at least three fifths.

The frozen summability bridge then supplies the nonzero t-product. The separately frozen convergence theorem carries the Multipliable assertion on this half-plane; the notation for the t-product alone is not being used as an existence claim.

This consequence advances one nonvanishing boundary in the OACTC 580/581 extraction ladder. It does not close the lower convergence strip, locate any germ zero, or imply the O-5 control statement or RH.

References

• Truth anchor: D5/S3/Analytic/EulerGerm/GermProductNonvanishingAboveThreeFifths.germ_local_factor_two_ne_zero_of_re_ge_three_fifths
• Truth anchor: D5/S3/Analytic/EulerGerm/GermProductNonvanishingAboveThreeFifths.germ_product_ne_zero_of_re_ge_three_fifths