Bruhat-Tits Tree as a Unifying Geometric Object

In the paper's words

Modern artificial intelligence, particularly in high-stakes domains, is hampered by a fundamental geometric mismatch: the reliance on continuous, “flat” Euclidean spaces to represent intrinsically hierarchical data. This incongruity leads to information loss and the proliferation of opaque “black box” models. This paper posits that the ultrametric geometry of p-adic numbers, which axiomatically encodes hierarchy, provides a more natural and powerful foundation for AI.

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

Numbers it reports

1. The p-adic embeddings preserve hierarchical structure with near-perfect fidelity (Spearman’s $\rho \approx 1.0$), significantly outperforming the highly distorted representations produced by Euclidean and hyperbolic proxy models.

   from A

2. Efficient methods for embedding into ultrametric spaces exist, but they are often treated as post-processing steps rather than a foundational aspect of the model architecture (Cohen-Addad et al., 2020).

   from 1.1 The Curse of

3. While classical methods are well-suited to analyzing static hierarchies (Murtagh & Adachi, 2017), they lack the dynamic efficiency required for real-time inference and exploration.

   from 1.3 Quantum Dynamics on the

4. The synthesis of p-adic geometry and quantum dynamics creates a system that is both structurally sound and computationally efficient, a combination that has been explored in adjacent theoretical domains (Quni-Gudzinas, 2025).

   from 1.3 Quantum Dynamics on the

5. This rapid, wave-like propagation is known as ballistic transport and is the source of the quadratic speedup (Hey et al., 2021).

   from 2.4 Quantum Walks: Theory

6. The p-adic geometry provides the static, structural backbone, ensuring that the relationships within the data are represented with perfect fidelity, as proposed by recent architectures (N’guessan, 2025).

   from 2.7 Synthesis:

How it is done

1. This work provides a simulation-based proof-of-concept for $O(D)$ navigation on a Bruhat-Tits tree for AI and offers a concrete pathway to address critical gaps in explainable AI (XAI) by creating models where the decision-making process is a geometrically interpretable path.

   from A

2. The existing literature, while rich in theoretical foundations, exhibits several critical gaps that prevent the realization of a fully integrated p-adic AI framework.

   from 1.4 Research Gap Analysis

3. This paper aims to provide a comprehensive, simulation-based proof-of-concept for a new class of AI models based on p-adic geometry and quantum dynamics.

   from 1.6 Scope and Contributions

4. This approach aligns with recent theoretical work that explores topological quantization and spectral filtration as a means of creating “prime-attentive” neural networks, which are designed to identify the most fundamental and indivisible features in a dataset (Quni-Gudzinas, 2025).

   from 2.3 Arithmetic Topology in

What would falsify it

1. We predict a growing interest in designing AI models that are “quantum-native,” meaning they are built from the ground up to leverage the principles of quantum mechanics, rather than simply adapting classical models to run on quantum computers.

   from 7.4 The Future of P-adic AI

Where the claim stops

1. A critical limitation of this study is its reliance on classical simulation of quantum dynamics.

   from 1.6 Scope and Contributions

2. We do not implement the proposed architecture on a physical quantum processing unit (QPU).

   from 1.6 Scope and Contributions

3. As such, our results on quantum speedup are theoretical and do not account for the effects of hardware noise or decoherence.

   from 1.6 Scope and Contributions

Where to read

1. A276 words · 1 min
2. 1.0 Introduction1,785 words · 8 min
3. 2.0 Theoretical Framework2,203 words · 10 min
4. 3.0 Methodology: The Q-PNA1,866 words · 8 min
5. 4.0 Results I:1,375 words · 6 min
6. 5.0 Results II:1,272 words · 6 min
7. 6.0 Discussion1,460 words · 6 min
8. 7.0 Conclusion802 words · 3 min
9. Appendices1,050 words · 5 min