Computational Topology and Formal Proofs of Discrete Non-Archimedean Relational Networks

In the paper's words

The fundamental reconciliation of quantum mechanics with macroscopic geometry requires a departure from continuous Archimedean manifolds, substituting them with discrete non-Archimedean topologies at the Planck scale. This foundational shift is contextualized by analyzing the Wheeler-DeWitt equation’s timeless configuration space, which reveals profound breakdowns in classical continuous descriptions. Our methodology employs strict discrete algebraic geometries, generating computational evidence via Python-simulated Bruhat-Tits trees and combinatorial node-counting algorithms.

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

Numbers it reports

1. However, despite this idealized baseline, the 5% modulation amplitude establishes a rigorous theoretical target well within future instrumental resolution limits.

   from 5.5 Simulating CMB

2. While the challenge of isolating 5% amplitude modulations from cosmic variance is steep, it is the most viable path to validating quantum gravity.

   from 7.6 Avenues for Future

3. Classical attempts to model black hole entropy frequently clash with continuous volume calculations, leading to infinite density paradoxes (Zúñiga-Galindo, 2023a).

   from 1.3 The

4. Our combinatorial calculations prove this one-to-one projection, confirming exact equivalence between interior relational depth and surface entropy (Jepsen, 2026).

   from 1.6 Holography in

5. The presented evidence proves that both holographic entropy and the arrow of time emerge statistically from unweighted network transversals (Mondal, 2025).

   from 1.7 Thesis Statement

6. Mathematical modeling confirms that string worldsheets efficiently map onto these trees, perfectly preserving conformal symmetries (Chen & Liu, 2021b).

   from 2.3 Bruhat-Tits Trees in

How it is done

1. The analytical results yield two profound physical validations: the Graph Laplacian natively generates a singular zero-mode modeling static Wheeler-DeWitt equilibrium, and the Area-to-Volume limit of tree networks natively converges to $(p-1)/p$, structurally guaranteeing the Bekenstein bound.

   from Addressing

2. Bekenstein Bound as Topological Constraint Information density within a physical system is strictly capped by topological geometry rather than external thermodynamic limits.

   from 1.3 The

3. However, recent proofs successfully mapping p-adic bounds to observable physics validated the approach’s ontological legitimacy, proving its physical relevance (Chen & Liu, 2021b).

   from 2.2 p-Adic Quantum Mechanics

4. Tensor networks applied to discrete geometries provide the most robust mathematical proof of the AdS/CFT holographic correspondence.

   from 2.4 Tensor Networks and AdS/CFT

What would falsify it

1. These discrete models accurately predict localized quantum clustering without relying on continuous background manifolds (Chen & Liu, 2021a).

   from 2.2 p-Adic Quantum Mechanics

2. Critics initially dismissed p-adic formulations as mathematical curiosities lacking clear physical mapping to macroscopic observable geometry.

   from 2.2 p-Adic Quantum Mechanics

3. However, recent proofs successfully mapping p-adic bounds to observable physics validated the approach’s ontological legitimacy, proving its physical relevance (Chen & Liu, 2021b).

   from 2.2 p-Adic Quantum Mechanics

Where the claim stops

1. Current Framework The primary limitation of this discrete topological framework is the exponential node scaling that forces computational models to operate on truncated, idealized subgraphs.

   from 7.5 Limitations of the

2. Traditional Archimedean continuous frameworks invariably fail at this boundary, producing non-physical singularities that demand a discrete reformulation (Hamber & Williams, 2011).

   from 1.1 The Archimedean

3. Historically, field theories have struggled to formalize this due to a reliance on background-dependent metrics that assume a pre-existing spatial void (Kauffman, 2021).

   from 1.2 Ontological

Where to read

1. Addressing229 words · 1 min
2. 1.0 Introduction and974 words · 4 min
3. 2.0 Literature867 words · 4 min
4. 3.0929 words · 4 min
5. 4.0985 words · 4 min
6. 5.0975 words · 4 min
7. 6.01,095 words · 5 min
8. 7.0 Conclusion and Future951 words · 4 min
9. Appendices227 words · 1 min