Computational Toolkit for p-Adic Spacetime

In the paper's words

The fundamental incompatibility between the continuous manifolds of general relativity and the discrete nature of quantum mechanics remains a central tension in theoretical physics. This manuscript develops and validates a computational toolkit designed to test theories of p-adic spacetime, which propose a pre-geometric substrate based on the Bruhat-Tits tree. We present a suite of Python-based methods including graph Laplacian generators for p-adic lattices, properly weighted Monte Carlo algorithms for simulating thermodynamic relaxation on trees, and a novel p-adic wavelet transform for analyzing boundary data.

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

Numbers it reports

1. FFT analysis of our dynamically generated noise successfully recovers non-harmonic peaks at 69Hz, 110Hz, and 161Hz, corresponding exactly to $p=2, 3, 5$ (Di Franco et al., 2022).

   from 5.7 Prime-Periodic Peak

2. Historically, the continuous-discrete tension has stymied quantum gravity, as standard quantization techniques applied to continuous metrics yield non-renormalizable ultraviolet divergences (Vladimirov & Volovich, 1989).

   from 1.1 Context: The

3. Time emerges as a Renormalization Group (RG) flow mapping to the cosmological scale factor along the depth of the tree (Hamber, Toriumi, & Williams, 2012).

   from 2.5 Discrete Wheeler-DeWitt

4. Entropy is mathematically linked to the prime zeta functions associated with the tree’s spectral properties (Jepsen, 2019).

   from 2.6 Thermodynamic Relaxation

5. The Laplacian is computed as $L = D - A$, utilizing scipy.sparse to manage matrix dimensions efficiently (Zúñiga-Galindo, 2024).

   from 3.2 Graph Laplacian

6. This accurately reflects the density of states and ensures valid thermodynamic entropy calculations (Huang, 2019).

   from 3.3 Monte Carlo Defect

How it is done

1. Empirical Links Despite the mathematical elegance of the Bruhat-Tits framework, the literature suffers from severe empirical and computational gaps.

   from 1.5 Gap Analysis: Missing

2. Equation Formulation To describe the kinematics of these defects, we define a dynamical operator on the tree that replaces the continuous d’Alembertian.

   from 2.4 The p-Adic Wave

3. Methodological Approach: Computational Bruhat-Tits Dynamics

   from 3.0

4. The tree is modeled as a directed graph class using sparse matrix representations to handle the exponential growth of nodes. significantly, our core tree-generation algorithm (detailed in Appendix B) has been verified for its scaling properties across different prime bases.

   from 3.1 Computational

What would falsify it

1. This mock data confirms that the wavelet pipeline can detect explicit, observable imprints characteristic of a p-adic boundary.

   from 4.4 CMB Mock Data Generation

2. This closes the gap by linking abstract Planck-scale topology directly to observable particle physics phenomena (Zabrodin, 1989).

   from 6.3 Mass Spectra and Defect

Where the claim stops

1. Pre-Geometric Paradigm The assumption of continuous spacetime manifolds, while highly successful in macroscopic general relativity, fundamentally breaks down at the Planck scale, necessitating a transition to a pre-geometric, discrete substrate.

   from 1.1 Context: The

2. Graph-based models approximate the continuous WdW constraint by treating the universe’s wavefunction as a distribution over discrete geometric configurations (Hamber, Toriumi, & Williams, 2012).

   from 1.3 The

3. The scope is restricted to the development and validation of these computational methods, using phenomenological models to demonstrate their efficacy.

   from 1.6 Research Questions and

Where to read

1. 1.0 Introduction671 words · 3 min
2. 2.0666 words · 3 min
3. 3.0627 words · 3 min
4. 4.0544 words · 2 min
5. 5.0 Results521 words · 2 min
6. 6.0473 words · 2 min
7. 7.0 Conclusion and Future423 words · 2 min
8. Appendices412 words · 2 min