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The Ultrametric Foundation: A Mathematical Thesis on Non-Archimedean Physics

DOI: 10.5281/zenodo.21993481
Published: 2026-08-18

The Ultrametric Foundation: A Mathematical Thesis on Non-Archimedean Physics

Record: v1.1.2 (2026-08-18) — self-DOI corrected (was 'pending'); see ERRATA.md

Author: Rowan Brad Quni-Gudzinas (ORCID: 0009-0002-4317-5604)

Date: 2026-07-12 | DOI: 10.5281/zenodo.21991899

License: QNFO Unified License Agreement (QNFO-ULA)

Version: 1.0.0 | Status: Complete (8 chapters)

Abstract

This thesis develops the complete mathematical foundation for non-Archimedean physics, unifying ultrametric topology, Bruhat-Tits buildings, Berkovich analytic geometry, Tate-Amice spectral analysis, Witt vector deformation theory, and topos theory into a single framework. The central result is that the category of ultrametric spaces with contraction mappings forms an elementary topos $\mathcal{E}_{\text{ult}}$ whose internal logic encodes non-Archimedean physical law. Eight chapters traverse from Ostrowski's classification of absolute values on $\mathbb{Q}$ through Bruhat-Tits geometric realizations, Berkovich analytic spectra, p-adic Fourier analysis, Witt vector deformation theory, the ultrametric topos construction, the Hasse local-global principle as descent, and physical applications unifying adelic quantum error correction, tree-depth time, Zitterbewegung cosmology, and adelic quantum field theory.

Keywords: ultrametric spaces, Bruhat-Tits buildings, Berkovich spaces, Amice transform, Witt vectors, topos theory, Hasse principle, adelic quantum computing

Table of Contents

  1. Ultrametric Spaces — Complete Classification — Ostrowski's Theorem, category $\mathbf{Ult}$, tensor product, Stone duality, p-adic numbers
  2. Bruhat-Tits Buildings as Geometric Realizations — BN-pairs, Coxeter complexes, affine buildings, $\mathrm{SL}(2,\mathbb{Q}_p)$ tree, classifying spaces, adelic QEC
  3. Berkovich Spaces and Analytic Geometry — Seminorm spectra, types of points, rigid/adic/comparison geometry, sheaf theory, Berkovich-Gelfand transform, tree correspondence
  4. Tate-Amice Spectral Analysis — Amice transform, Mahler expansion, p-adic wavelets, multiresolution analysis, non-Archimedean spectral theorem, Mahler compression as computational primitive
  5. Witt Vectors and Deformation Theory — Witt polynomials, Teichmüller lift, deformation theory, Greenberg transform, Green functors, crystalline cohomology, prismatic cohomology
  6. The Ultrametric Topos — Topos construction, internal logic (Boolean with controlled AC failure), Pontryagin duality morphism, quantum logic realization, Isham-Döring program
  7. Hasse Local-Global Principle — Classical Hasse-Minkowski, descent reformulation, Brauer-Manin obstruction, adelic cohomology, Grothendieck topology, physical examples
  8. Physical Applications — Capstone — Kepler adelic QEC, tree-depth time, Zitterbewegung cosmology, adelic QFT, Laws of Form isomorphism

Individual chapter files available in the source repository at ultrametric-foundation-ch{1-8}.md.