ONE AXIOM : The Critical Ridge : Phase Coherence as a Structural Invariant of ζ(s)

7B — The Critical Ridge: Phase Coherence as a Structural Invariant of $\zeta(s)$

Description:

Overview

This paper presents a formal structural establishment of the Geometric Riemann Hypothesis (G-RH) within the ONE AXIOM framework. Moving beyond traditional analytic number theory, document 7B identifies the distribution of Riemann zeros as a structural necessity of holographic decompression. By employing a Dual-Track Methodology, the work bridges Lens 2 (Geometric/Classical) and Lens 4 (Ontological), revealing RH as a compatibility condition between the VOID ($\pi_4$) and the relational layer ($\pi_6$).

Key Contributions

• The Bridge Theorem: The central result of the paper, establishing that G-RH implies the Classical Riemann Hypothesis. The theorem demonstrates that phase pinning at $\Phi \in \{0, \pm\pi\}$ and the exclusion of off-axis quadruplets are mandatory for ontological coherence.

• Possibility Selection Principle (PSP): Application of the 51:13 structural filter derived from Fano-K7 duality. The paper demonstrates that 13 out of 64 phase sectors are "FORBIDDEN," preventing any off-axis zero formation.

• Statistical Validation: A rigorous analysis of the first 400 Riemann zeros shows an overwhelming fit for the ONE AXIOM model ($\chi^2 = 713.22$) with a Bayes Factor ($BF > 10^{154}$), ruling out the null hypothesis of random distribution.

• Independent Experimental Validation: Integration of recent findings by Wei et al. (arXiv:2511.11199), providing external empirical support for the structural predictions of the model.

• Quantum Gap Analysis: The analysis reveals a 90-point quantum gap: 100% of on-critical zeros occupy ALLOWED sectors, versus only 10.2% of off-critical points. This disparity confirms the "Critical Ridge" as a preferred state of ontological stability.

Methodology & Formalism

The work introduces an "Epistemological Switch," shifting from numerical verification to structural recognition. It contains 8 formal theorems (Section 8) that define the Ontological Coherence Field (OCF) and the mechanism of phase-locking. The paper formalizes the condition $\sigma = 1/2 \iff \Phi \in \{0, \pm\pi\}$ on the critical line, linking Hardy’s Z-function to the $q^ = 3/2$* Tsallis critical exponent at Layer $\pi_{5.5}$.

Target Audience

Researchers in complex analysis, quantum chaos, and theoretical physics interested in non-extensive statistics, holographic principles, and the structural foundations of mathematics.