Geometry and Spectral Dynamics of Coherence Graphs:

We study dynamically generated weighted graphs whose geometry and spectral structure emerge from coherence-based update rules on edge weights. An effective metric is induced by negative log weights on the stabilized graph, yielding an intrinsic geometry without an external manifold. We define a periodic-orbit zeta function built from primitive nonbacktracking cycles with real-valued lengths and relate its analytic structure to the spectrum of weighted nonbacktracking (Hashimoto-type) operators. Motivated by the Ihara–Bass theory for regular graphs, we propose a falsifiable RH-analog conjecture: coherence-optimal integration—characterized by the absence of persistent spectral outliers—corresponds to alignment of nontrivial zeta zeros on a critical locus determined by the spectral radius of the associated nonbacktracking operator. A minimal Hebbian-style toy dynamic with normalization is introduced, and numerical experiments demonstrate convergence to a stabilized coherence matrix, grounding the construction in explicit dynamics. The results are purely dynamical and geometric in nature and make no claims regarding arithmetic zeta functions or the classical Riemann Hypothesis.