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Spectral Dynamics on Bruhat-Tits Trees: A Deterministic p-adic Framework for Primality Testing

Rowan Brad Quni-Gudzinas

Published Feb 13, 2026• 0 citations
Publisher

Abstract

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. Our methodology combines a rigorous construction of the graph Laplacian for these spaces with a continuous-time quantum walk simulation. 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 $\approx 11.97$). This binary dynamic behavior—localization versus delocalization—serves as a deterministic signal for primality. Furthermore, multi-base triangulation across $p=2, 3, 5$ yields a predictive correlation of $R^2 = 0.998$, confirming the robustness of the underlying metric. These findings establish a framework for a novel class of deterministic primality tests, relying on dynamic signatures rather than static spectral properties.

DOI
10.5281/zenodo.18629518

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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. Our methodology combines a rigorous construction of the graph Laplacian for these spaces with a continuous-time quantum walk simulation. Analysis reveals that while both structures exhibit a robust, non-zero spectral gap (λ2≈0.0968\lambda_2 \approx 0.0968λ2​≈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 ≈11.97\approx 11.97≈11.97). This binary dynamic behavior—localization versus delocalization—serves as a deterministic signal for primality. Furthermore, multi-base triangulation across p=2,3,5p=2, 3, 5p=2,3,5 yields a predictive correlation of R2=0.998R^2 = 0.998R2=0.998, confirming the robustness of the underlying metric. These findings establish a framework for a novel class of deterministic primality tests, relying on dynamic signatures rather than static spectral properties.
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The prevailing quantum computing paradigm is fundamentally constrained by the continuous analog fragility inherent to Archimedean spaces, resulting in uncontrolled linear error accumulation that demands massive active correction overheads. To address this structural bottleneck, we propose a transition to a non-Archimedean state space modeled on the p-adic numbers and their graph-theoretic realization, the Bruhat-Tits tree. To validate this non-Archimedean architecture without physical p-adic hardware substrates, we developed a comprehensive software emulation utilizing Bounded Algorithmic Number (BAN) arithmetic logic units to process exact ultrametric valuations. We simulated the compilation of continuous logic gates into discrete tree automorphisms, mapping standard unitary operators to vertex shifts and branch permutations on a p=2 Cayley graph. Furthermore, stochastic Ohmic and burst noise injections were modeled to test the error-filtering properties of the strong triangle inequality dynamically, while van der Put Neural Networks (v-PuNNs) were deployed as read-out trackers to prevent topological distortion. Our empirical simulation results demonstrate that non-Archimedean architectures natively suppress linear error accumulation, with variance saturating flatly at local cluster boundaries, confirming passive geometric fault tolerance. These findings carry implications for the future of post-NISQ hardware, formally bridging holographic tensor network theory with applied digital emulation to outline a scalable computational substrate.
The Bruhat-Tits tree T_p is proposed as a geometric substrate for quantum error-correcting codes. The defining property of ultrametric geometry replaces continuous error accumulation with discrete redundancy steps. This topological reframing connects p-adic mathematical physics, quantum error correction theory, and quantum Darwinism. The central conjecture is that the redundancy-fidelity tradeoff, parameterized on a real quantum processor, may exhibit a staircase structure (ultrametric regime) rather than a smooth curve (Archimedean limit), and that this structure — if detected — would constitute a new QEC design parameter. An experimental protocol is presented for discriminating between these regimes on existing quantum processors. All claims are graded with certainty labels; the paper acknowledges that the experimental program is the path to evidence. [v0.2 — literature search completed 2026-08-06: prior holographic-code work (Marcolli 2018) acknowledged; novelty claim refined to device-level QEC with staircase detection.]
The Bruhat-Tits tree provides an ultrametric substrate for quantum error-correcting codes whose information topology differs from the Archimedean case in ways that may revise the redundancy-fidelity tradeoff established for quantum Darwinism. Two limiting regimes are identified: the large-p limit, in which the ultrametric bound converges to the Archimedean bound, and the deep ultrametric regime of small primes, in which discrete redundancy steps create windows where quantum error correction and classical objectivity may coexist. An experimental protocol is proposed to detect the staircase structure on real quantum processors. All claims are graded with epistemic certainty labels and subjected to an internal red-team audit. Licensed under QNFO Unified License Agreement (QNFO-ULA).
Continuous-time quantum walks (CTQWs) exhibit localization phenomena that differ fundamentally from their classical counterparts, yet the precise relationship between network structure, spectral degeneracy, and confined dynamics remains incompletely understood. In this work, we present a complete analytical characterization of localization in CTQWs on two highly symmetric graph families: barbell graphs and star-of-cliques graphs. These networks combine pronounced spectral degeneracy with modular structure, enabling exact diagonalization and explicit computation of both eigenstate and dynamical inverse participation ratios (IPRs). Our analysis reveals that localization is governed by the interplay between degenerate subspaces, which generate families of confined modes, and hybridization between invariant subspaces, which redistributes spectral weight. Notably, the dynamical IPR can exceed expectations based solely on eigenstate IPRs, demonstrating that coherent superposition within degenerate eigenspaces enhances confinement. By connecting IPR values to the effective number of vertices visited, we provide a structural diagnostic for predicting quantum transport outcomes in modular networks, establishing that connectivity alone can determine where and how strongly a quantum walk localizes.
The standard mathematical foundation of quantum computing—Archimedean (Euclidean) geometry—assumes that space is continuous, distances add linearly, and small errors can accumulate into large ones. This assumption creates a fundamental bottleneck: active quantum error correction generates heat that scales faster than cryogenic cooling capacity, producing a 20,000×20{,}000\times20,000× gap between required and available cooling at millikelvin temperatures. We propose an alternative: encoding quantum states on Bruhat–Tits trees, the geometric realization of ppp-adic (ultrametric) spaces, where the strong triangle inequality d(x,z)≤max⁡{d(x,y),d(y,z)}d(x,z) \leq \max\{d(x,y), d(y,z)\}d(x,z)≤max{d(x,y),d(y,z)} geometrically confines errors to their local branches. We present a computational validation suite (classical simulation) demonstrating this mechanism. Our results show that tree-encoded circuits produced zero observed logical errors in 500 trials at depths d≥3d \geq 3d≥3 for physical error rates up to perr=0.40p_{\text{err}} = 0.40perr​=0.40, while equivalent flat (Archimedean) encodings failed with logical error rates up to 0.1520.1520.152. The energy barrier protecting logical states scales as Ebarrier(d)=2dE_{\text{barrier}}(d) = 2^dEbarrier​(d)=2d, confirmed exhaustively for d=2,3d = 2, 3d=2,3 and computed analytically to d=10d = 10d=10. The strong triangle inequality is verified with zero violations in 15,00015{,}00015,000 random trials across primes p=2,3,5p = 2, 3, 5p=2,3,5. These results provide computational evidence for a geometric threshold theorem in ultrametric quantum error correction and suggest a path toward passive fault tolerance—error suppression as a property of hardware geometry rather than active measurement protocols. The simulation uses classical bits with majority-vote decoding; full quantum validation with superposition and entanglement remains future work.