P-adic Spin, Information, and Ultrametric Internal Quantum Numbers

In the paper's words

P-adic completions of Q encode discrete quantum numbers structurally via Bruhat-Tits geometry: p=2 gives spin, p=3 gives color charge. The Z2 invariant distinguishes Dirac from Majorana.

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

Numbers it reports

1. The classification achieves 83% accuracy with 100% lemma-level agreement across 4 code families.

   from 4.3 The Kodaira-Néron Classification and Code Structure

How it is done

1. Quantum mechanics presents a striking dualism: Dynamics are Archimedean: wavefunctions evolve continuously in time via the Schrödinger equation, operators have continuous spectra (position, momentum), and the intermediate value theorem governs transitions.

   from 1.1 The Problem

2. By Ostrowski's theorem (1916), the only non-trivial completions of $\mathbb{Q}$ are: $\mathbb{R}$ (the Archimedean completion, the $\infty$-place) — connected, ordered, the domain of dynamics and causality. $\mathbb{Q}_p$ for each prime p (the ultrametric completions) — totally disconnected, tree-structured, the domain of discrete quantum structure and information.

   from 1.2 The Thesis

3. Theorem 2.1 (Ultrametric = Hierarchical).

   from 2.2 The Ultrametric Inequality

4. The spin-statistics theorem in 3+1 dimensions rests on two facts: $\pi 1(SO(3))$ = $\mathbb{Z}_2$ — the rotation group is not simply connected; its double cover is SU(2).

   from 3.1 Spin Is Topological, Not Metric

What would falsify it

1. The adelic framework makes specific, falsifiable predictions: Gromov $\delta = 0$ for spin/ZBW transitions.

   from 5.3 Falsifiable Predictions

2. Experimental validation: Execute the falsifiable predictions of §5.3 — Gromov $\delta$ measurement, ZBW $\mathbb{Z}_2$ invariant detection, and p-adic code stratification.

   from 7.3 Open Questions

Where the claim stops

1. It does not explain why the continuum formalism admits discrete structure at all.

   from 1.1 The Problem

2. It does not depend on the metric on SO(3) — only on the underlying topological space.

   from 3.1 Spin Is Topological, Not Metric

3. The mutual singularity of the topologies means that any error process representable as an Archimedean perturbation (which includes all standard decoherence mechanisms — thermal, electromagnetic, vibrational) cannot affect information encoded in p-adic fixed points.

   from 5.2 Mutual Singularity and Adelic QEC

Where to read

1. 1. Introduction: The Puzzle of Discrete Quantum Numbers265 words · 1 min
2. 2. P-adic Numbers as Information Trees356 words · 2 min
3. 3. Spin from the 2-adic Place427 words · 2 min
4. 4. Internal Quantum Numbers as Bruhat-Tits Vertex Labels351 words · 2 min
5. 5. Time as Archimedean, Information as P-adic354 words · 2 min
6. 6. Relation to String Theory's Extra Dimensions243 words · 1 min
7. 7. Conclusion276 words · 1 min