Spectral Analysis of Anomalous Diffusion on p-Adic Fractals

In the paper's words

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.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

No sentence of this paper reports a figure with a unit or a quantity word, so none is shown.

How it is done

1. Despite the richness of the field, several critical gaps prevent a unified theory.

   from 1.5 Gap Analysis

2. To bridge these gaps, we implement a comprehensive random walk simulation on constructed graph topologies representing both the standard integer line and p-adic fields.

   from 1.6 Methodological Approach

3. By transforming the “time” variable of the search algorithm into a frequency domain, the “knots” of divisibility—which appear as random gaps in linear time—might align into resonant peaks.

   from 2.6 Geometric Factorization

4. Protocol To simulate p-adic diffusion, we constructed graph topologies representing the bulk geometry of $\mathbb{Q}_p$.

   from 3.1 Graph Construction

What would falsify it

1. We treat them here as theoretical probes to be tested against standard diffusion models.

   from 1.4 Radical

Where the claim stops

1. Simulation We must acknowledge that our finite tree depth ($D=10$) only approximates the infinite p-adic limit.

   from 6.5 Limitations of the

2. However, a critical gap persists in physically realizing this substrate: how can a system be simultaneously discrete (like primes) and possess the continuous spectral properties required by quantum mechanics?

   from 1.1 Contextual Landscape

3. Standard diffusion fails here; walkers get trapped in the deep branches of the divisibility tree (Torres & Das, 2024).

   from 1.2 The Discrete-Continuous

Where to read

1. Reconciling271 words · 1 min
2. 1.0 Introduction806 words · 4 min
3. 2.0669 words · 3 min
4. 3.0542 words · 2 min
5. 4.0 Results I:532 words · 2 min
6. 5.0573 words · 2 min
7. 6.0471 words · 2 min
8. 7.0 Conclusion & Future320 words · 1 min
9. Appendices1,494 words · 6 min