Spectral Dynamics on Bruhat-Tits Trees

In the paper's words

The probabilistic nature of prevalent primality tests, such as Miller-Rabin, presents a fundamental limitation where certainty is achieved only asymptotically. This study proposes a deterministic alternative rooted in the non-Archimedean geometry of p-adic numbers, specifically leveraging the spectral and dynamic properties of quantum walks on Bruhat-Tits trees. By modeling prime and composite numbers as distinct topological structures—regular trees for primes and Cartesian product graphs for composites—we demonstrate that their quantum dynamic signatures are fundamentally different.

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

Numbers it reports

1. By setting a detection threshold of $P \gt 0.10$, the algorithm successfully identified the target factor nodes in 100% of the simulated trials for the $p=2$ tree.

   from 5.3 Factor Identification Accuracy |

2. Analysis reveals that while both structures exhibit a robust, non-zero spectral gap ($\lambda_2 \approx 0.0968$), their dynamic responses diverge: quantum walks on prime trees exhibit localization, whereas walks on composite product spaces lead to near-maximal delocalization across the graph (Participation Ratio $\approx 11.97$).

   from Abstract

3. Additionally, the observation of a localization probability $P \gt 0.5$ at the resonance time serves as confirmation of the ballistic transport mechanism’s efficacy for factor identification.

   from 3.7 Validation Strategy |

4. However, we observed a slight difference in the degeneracy ratio (a measure of unique eigenvalues), with the prime tree showing a ratio of 0.60 and the composite product graph showing 0.58, hinting at a more complex spectral structure for composites.

   from 4.2 Spectral Gaps and Prime Factors |

5. The time complexity of the quantum evolution is $O(\log N)$, confirming an exponential speedup over classical random walks (Childs, 2002).

   from 5.4 Scaling with Integer Size |

How it is done

1. Keywords : p-adic, Quantum Mechanics, Primality Testing, Quantum Walk, Bruhat-Tits Tree, Spectral Graph Theory, Localization, Deterministic Algorithm, Non-Archimedean Geometry, Hamiltonian Simulation, Computational Number Theory

   from Abstract

2. While efficient, these methods do not provide mathematical proof of primality; instead, they offer a high degree of confidence that diminishes with the complexity of the number.

   from 1.1 The Limits of Probabilistic Primality Testing |

3. To achieve these objectives, we employ a computational simulation of a tight-binding Hamiltonian on finite p-adic graphs.

   from 1.5 Methodological Approach |

4. By providing a method grounded in the fundamental geometry of numbers, it could offer insights into the structure of primes that are inaccessible to probabilistic methods.

   from 1.6 Significance & Impact |

What would falsify it

1. The revised, rigorous simulation using product graphs falsified this hypothesis.

   from 4.2 Spectral Gaps and Prime Factors |

2. The ability to predict factors using geometric resonance implies that the information is present in the system all along, encoded in the non-local correlations of the p-adic metric.

   from 6.1 Revisiting P-adic Causality |

Where the claim stops

1. The probabilistic nature of prevalent primality tests, such as Miller-Rabin, presents a fundamental limitation where certainty is achieved only asymptotically.

   from Abstract

2. While efficient, these methods do not provide mathematical proof of primality; instead, they offer a high degree of confidence that diminishes with the complexity of the number.

   from 1.1 The Limits of Probabilistic Primality Testing |

3. Similarly, while ballistic transport on hierarchical graphs has been observed (Boettcher, 2020), its application has been largely restricted to abstract transport phenomena rather than number-theoretic computation.

   from 1.3 Historical Context & Literature Gaps |

Where to read

1. A22 words · 1 min
2. Abstract217 words · 1 min
3. 1.0 Introduction |742 words · 3 min
4. 2.0 Theoretical Framework: P-adic Physics & Graph Theory |1,027 words · 4 min
5. 3.0 Methodology: Hamiltonian Construction |883 words · 4 min
6. 4.0 Results I: Spectral Analysis of Bruhat-Tits Trees |821 words · 4 min
7. 5.0 Results II: Ballistic Dynamics & Factorization |778 words · 3 min
8. 6.0 Discussion: Determinism, Entropy, & Cryptography |554 words · 2 min
9. 7.0 Conclusion |173 words · 1 min
10. Appendices608 words · 3 min