Pure Time Theory – Chapter V - A Spectral–Geometric Route to the Riemann Hypothesis
Year: 2025
Type: article
Abstract: This article presents the definitive resolution of the Riemann Hypothesis (RH) within the Pure Time Theory (PTT) framework, replacing and superseding my earlier preprint (*The Riemann Hypothesis as a Derived Theorem in Pure Time Theory*, v24). The v24 version formulated RH as a theorem inside an axiomatic system T (axioms T1–T5), showing that injectivity constraints on temporal phases force all non-trivial zeros of zeta(s) onto the critical line. That work was an essential first step, but it remained abstract and axiomatic. The present article (from Intuition to Genesis Chapter V v21) completes the loop opened by Chapters I–III of the Genesis of PTT. Building on the zero-induced geometry developed earlier, the proof now proceeds in a purely analytic and geometric way: 1. From an optical slice we construct a compact, odd boundary datum W. 2. Using Paley–Wiener synthesis, we build entire functions consistent with W. 3. A canonical separation isolates the Archimedean background, on-line zeros, and off-line factors. 4. Hardy–Smirnov rigidity enforces constancy of the comparison quotient. 5. Injectivity and the intrinsic flux–neutral gauge eliminate the off-line factor, proving that all zeros lie on Re(s)=1/2. This new derivation abandons the external axiomatic container and works directly with the geometry encoded by the zeta zeros themselves. It yields a fully self-contained, non-circular proof of RH that is compatible with classical analytic number theory while rooted in the modal architecture of PTT. The transition marks the maturation of PTT from a speculative axiomatic program into a concrete, structural demonstration that RH is a necessary consequence of zero-induced geometry.
Author: Allou, Essam
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6968577326
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