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Ultrametric Information Geometry: From p-Adic Spaces to Quantum Error Correction

Rowan Brad Quni-Gudzinas

Published Jul 5, 2026• 0 citations
Publisher

Abstract

Ultrametric spaces provide a natural geometry for hierarchically structured information. This paper synthesizes four independent research programs: Murtagh's data embedding in ultrametric spaces, Heydeman's holographic QEC on Bruhat-Tits trees, Palmer's p-adic quantum foundations, and Aniello's p-adic Hilbert space formalism. QNFO contributions: D=4 embedding theorem, adelic QEC, CMB analysis.

DOI
10.5281/zenodo.21204114

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Ultrametric spaces provide a natural geometry for hierarchically structured information. This paper synthesizes four independent research programs: Murtagh's data embedding in ultrametric spaces, Heydeman's holographic QEC on Bruhat-Tits trees, Palmer's p-adic quantum foundations, and Aniello's p-adic Hilbert space formalism. QNFO contributions: D=4 embedding theorem, adelic QEC, CMB analysis.
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).
Six papers, one thesis: **Physics is adelic.** The Archimedean (real-number) description is the ∞\infty∞-place readout of a richer ultrametric structure defined over the p-adic completions of Q\mathbb{Q}Q. Ostrowski's theorem — which classifies all non-trivial completions of the rational numbers as R\mathbb{R}R and Qp\mathbb{Q}_pQp​ for each prime ppp — is the hidden organizing principle behind quantum measurement, quantum error correction, and the structure of spacetime at the Compton scale. We present the unified framework: Zitterbewegung (ZBW) is the physical manifestation of the p-adic channel of the adelic Dirac equation; Majorana zero modes are Bruhat-Tits fixed points encoding adelic topological charge; and adelic quantum error correction replaces active QEC codes with number-theoretic protection. This synthesis paper connects all six companion publications into a coherent research program with specific, falsifiable predictions. **Keywords:** Adelic physics, Ostrowski's theorem, Zitterbewegung, Bruhat-Tits tree, p-adic quantum mechanics, topological quantum computing, grand synthesis ---