Spectral Analysis of Anomalous Diffusion on p-Adic Fractals: Reconciling Riemannian Geometry with Discrete Arithmetic via Geometric Resonances

The reconcilement of continuous Riemannian geometry with the discrete arithmetic of the integers remains one of the most profound challenges in mathematical physics, particularly regarding the spectral interpretation of the Riemann zeros (Berry & Keating, 1999). While standard diffusion on Archimedean manifolds is well-understood, the “knots” of prime gaps create topological obstructions best modeled by non-Archimedean, p-adic geometry. This study simulates anomalous diffusion on Bruhat-Tits trees to quantify these obstructions, establishing a computational framework to test radical geometric factorization hypotheses. We implemented Monte Carlo random walks on $p$-regular trees ($p=2, 3$) and extracted effective transient dimensions ($d_{eff}$) from return probability decays, contrasting these with baseline integer diffusion. Our analysis reveals a stark dimensionality paradox: while the integer line $\mathbb{Z}$ exhibits standard diffusion with $d_s \approx 1.0$, the $p$-adic bulk manifests as a high-dimensional fractal trap with effective transient dimensions $d_{eff} \approx 6.2$ for $p=2$ and $d_{eff} \approx 7.9$ for $p=3$. These values confirm that the “smooth” p-adic ring $\mathbb{Z}_p$ is dynamically monstrous, trapping information in exponential branches. Furthermore, spectral analysis of the finite tree Laplacian indicates a highly degenerate spectrum distinct from the Gaussian Unitary Ensemble (GUE) chaos expected of the Riemann zeros, confirming that pure ultrametricity is insufficient to capture the “Arithmetic Chaos” without broken symmetry. These findings support the “Geometric Factorization” hypothesis only insofar as they identify the deep topological entanglements—the “knots”—that a geometric unfolding must resolve.