The Riemann Hypothesis via a Lyapunov Dynamic Cascade in the Explicit Formula

Priest, Eliahi

doi:10.5281/zenodo.17981485

Published December 19, 2025 | Version 8.0 - Submission Ready Edition.

Preprint Open

The Riemann Hypothesis via a Lyapunov Dynamic Cascade in the Explicit Formula

Priest, Eliahi (Rights holder)

This manuscript presents a Clay-compliant framework and claims an unconditional proof of the Riemann Hypothesis using a Lyapunov Dynamic Cascade within the classical explicit-formula setting. The method studies the horizontal behaviour of the completed zeta function through admissible spatial test kernels and time-windowed energy measurements, without altering or deforming ζ or ξ at any stage.

The central mechanism is a fixed-T contradiction. If a nontrivial zero lay off the critical line, its local behaviour would force a non-integrable cusp in a horizontal Lyapunov energy, and hence divergence of the corresponding Gaussian-windowed energy for every fixed measurement scale T>0. Independently, an unconditional bound derived from the Guinand–Weil explicit formula yields a finite polylogarithmic envelope for the same windowed quantity, with all dependence on T made explicit. These two statements cannot simultaneously hold unless all nontrivial zeros lie on the critical line.

The analysis relies only on classical tools from analytic number theory: explicit-formula identities, vertical-line estimates, unconditional unit-band zero counting, Poisson-kernel representations of the zero block, and quadratic-form/Friedrichs theory for the spatial energy. No deformation techniques, spectral ansätze, or unproved hypotheses are used. All regulators act solely as external test weights and are removed in justified limits, following a strict “measure, not modify” principle.

This Submission-Ready Cascade Edition consolidates earlier developments by isolating a fully non-circular, fixed-T framework built entirely from unconditional estimates and locally quantified behaviour near a hypothetical off-critical-line zero. It is submitted in the spirit of mathematical clarity, adhering to the principle of proving one precise claim with complete internal justification.

Dedicated to the hardworking and remarkable people of Africa, where a substantial portion of this work was completed.

Keywords: Riemann Hypothesis; explicit formula; Lyapunov method; analytic number theory; windowed energy; quadratic forms; divergence–versus–bound contradiction.
MSC 2020: Primary 11M06, 11M26; Secondary 47A10, 47B25.

Technical info

Additional Notes / Version History

Version 8 (Submission Ready Edition) - 19 December 2025

Version 8 finalises the proof for submission. It incorporates the corrected handling of the zero-block frequency envelope, including an explicit domination argument for the sup⁡Δ regime, completes the Dirichlet–Euler block correction via a Parseval/frame estimate, and unifies all notation and cross-references. All equations are numbered. No new hypotheses or results are introduced relative to Version 7; this release makes explicit the minimal justifications required for referee-level scrutiny.

Version 7 (Cascade Edition) - 7 December 2025

This version introduces the Lyapunov Dynamic Cascade and is the first formulation to establish a fully T-controlled framework. The proof now uses an explicit contradiction at a fixed measurement scale: any hypothetical off-line zero forces a local cusp in the horizontal energy that diverges at that same fixed T, while the global explicit-formula bound provides finite T-controlled behaviour with explicit dependence on T.

Because the argument is carried out at a single fixed value of T, no uniformity in T or global integrability assumptions are required. This eliminates the global L² circularity present in Version 6 and earlier drafts.

The method is strictly “measure, not modify”: kernels and windows act only as external test operators, and all limit operations are justified from standard analytic tools. The abstract has been rewritten to reflect this cascade-based, T-controlled structure, which is the first Clay-standard and fully non-circular version of the argument.

Versions 3–6 (6 Oct 2025)

Summary of technical revisions

Versions 3–6 focused on resolving outstanding referee-level issues in the explicit-formula analysis. These revisions repaired the handling of individual EF blocks, clarified cross-term structure, and ensured logical independence of steps. However, they did not provide a T-controlled global bound and did not resolve the global L² circularity that remained in the zero-flux formulation. That issue is addressed only in Version 7.

Prime-block reconstruction

Earlier drafts relied on a time-domain inequality that was not valid across all regimes. Versions 3–6 replaced this with a mean-square argument: the Gaussian window was inserted into the linear explicit formula, Plancherel was applied with a Gaussian multiplier, and coefficient control was obtained via Mellin-shadow decay, Montgomery–Vaughan–type estimates, and Schur bounds. This fixed an inadmissible inequality but did not yet produce the final T-controlled EF envelope.

Gamma and zero blocks

The gamma contribution was reorganised so that its dominant growth terms cancel before squaring.
The zero block was reformulated through a Poisson-kernel representation and estimated via a band-limited Schur argument together with the unconditional unit-band zero count. These steps repaired correctness and clarity but remained within the zero-flux framing.

Parseval bridge and cross-term clarification

The relationship between coefficient norms and the horizontal energy was written explicitly, removing earlier ambiguity and ensuring the assembly of the EF bound followed a transparent chain of identities.

Logical independence

All integrals were placed on firm footing using Tonelli’s theorem, and constants were declared independent of window parameters where applicable. These steps corrected local technical issues but did not change the overall structure of the proof.

Label and reference unification

The explicit-formula bound was consolidated into a single proposition, and related appendices were refined to reflect the corrected linear identities and symbol-decay mechanisms.

Abstract

Unchanged across Versions 3–6; now superseded in Version 7.

Result

Versions 3–6 resolved technical correctness issues within the explicit-formula derivations but retained the global L² framework and the zero-flux interpretation. The deeper structural obstruction, the reliance on a global integrability regime equivalent to the conclusion, remained unresolved.

Version 7 replaces the entire mechanism with a fixed-T Lyapunov cascade built solely from unconditional explicit-formula estimates and local cusp analysis. This produces the first non-circular, Clay-standard formulation of the proof.

Earlier versions are preserved only to document the evolution of ideas.

Technical info

Version 2 note (29 Sep 2025). We clarified the explicit‑formula assembly in §4.8: the time window is inserted at the linear level and the frame coefficients are summed after the linear cancellation of the ℜ(Γ′/Γ)\Re(\Gamma'/\Gamma)ℜ(Γ′/Γ) main term. This removes the only place where a three‑squares estimate could introduce a spurious log⁡2T\log^2 Tlog2T envelope for the gamma block. The global T‑uniform bound is unchanged; no results, lemmas, or appendices were altered, this is a presentation‑level clarification.

Notes

Author’s Note

This manuscript arises from independent research carried out outside institutional mathematics. Its methods were developed through long study of classical analytic number theory and the explicit formula, combined with a systems-based perspective shaped by work in the applied and physical sciences. Although I am not a professional number theorist by training, every argument presented here belongs fully to the classical analytic tradition: it is elementary in the technical sense, grounded in standard identities, and requires no modification of the zeta or xi functions at any stage.

Large language models were used only as tools for editing, structuring, and maintaining consistency of exposition. They did not generate the mathematical ideas, the strategy, or the proofs contained in this work. All mathematical responsibility, the correctness of the reasoning and any errors that may remain, rests entirely with the author.

Mathematics is a cumulative human endeavour. Its progress depends on the willingness of individuals, whether inside or outside the academy, to undertake careful study and contribute work to be openly examined by the community. This manuscript is submitted in that spirit. The claim it makes touches one of the central questions in analytic number theory; it is therefore offered to the world’s mathematicians with full respect for the standards of scrutiny such a claim must withstand.

The manuscript is submitted with respect for the long tradition of work surrounding the Riemann Hypothesis and with full openness to rigorous verification. The mathematical community’s evaluation will determine its place within that tradition.

Notes

About the Author

Professor Eliahi Priest is the Founder and Chief Executive Officer of The Priest Group Pty Ltd, operating in Australia and across Africa. His professional work spans scientific research, ethical finance, and governance, with a focus on transparent and auditable sovereign systems.

From 2005 to 2007, Priest was mentored by Professor Robert Pope of the Science Art Research Centre (Australia), a UNESCO-recognised scientist whose work on symmetry, aesthetics and dynamical balance contributed to the conceptual foundations later developed into the Equation of Relational Unity. Building on this tradition, Priest formulated the Kairos Codex, a systems-based operator framework intended to examine major open problems in mathematics and theoretical physics.

Priest has advised at the presidential level in the Democratic Republic of Congo and Kenya on ethical sovereign-finance initiatives, has addressed the UN-mandated World Peace Congress, and founded Unified Field Science & Emergent Intelligence (UFSEI) to advance interdisciplinary research at the intersection of mathematics, physics, and ethical systems design.

This manuscript on the Riemann Hypothesis represents the central mathematical contribution of that programme: an effort to demonstrate, in a single rigorous case, how coherence-based operator methods may illuminate long-standing questions in analytic number theory.

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