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ZENODO
Preprint . 2026
License: CC BY
Data sources: ZENODO
ZENODO
Preprint . 2026
License: CC BY
Data sources: Datacite
ZENODO
Preprint . 2026
License: CC BY
Data sources: Datacite
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A boundary product–certificate method for the Riemann zeta function: far-field zero-freeness, near-field energy barriers, and conditional closure of the Riemann Hypothesis

Authors: Washburn, Jonathan;

A boundary product–certificate method for the Riemann zeta function: far-field zero-freeness, near-field energy barriers, and conditional closure of the Riemann Hypothesis

Abstract

We prove that all nontrivial zeros of the Riemann zeta function with Re(s) >= 0.6 lie on the critical line (unconditional), and establish an effective zero-free barrier for the remaining strip with astronomical protection heights. Far-field (Re(s) >= 0.6): Unconditionally zero-free. The arithmetic Cayley field Θ is Schur (|Θ| . Combining the barrier with the full Carleson bound yields an explicit, computable protection height Tsafe(eta) (see (3)); for example, at depth eta = 0.1 one obtains Tsafe approx 10^74, and at depth eta = 0.01 one obtains Tsafe >~ 10^8800. Main Result (Theorem 183). (a) (Unconditional) The Riemann zeta function has no zeros with Re(s) >= 0.6. (b) (Effective) No zeros with 1/2 < Re(s) < 0.6 exist at height |t| <= Tsafe(eta). For eta = 0.01, the protection height is on the order of 10^8800—far beyond any conceivable computation. Complete closure via Recognition Science. The remaining gap—bounding the zeros contribution uniformly in height—is closed within the Recognition Science axiomatic framework. Under the Nyquist Coverage Bound (Axiom T7, a theorem of the deeper axioms T2 and T6), the log T growth is eliminated, yielding the full Riemann Hypothesis (Corollary 84).

Keywords

Recognition Science, Pick matrices, Riemann zeta function, riemann hypothesis, riemann, zero-free region, Schur functions

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
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