p‑Adic Ancestry Model: A Deterministic Ultrametric Ontology for Quantum Correlations

In the paper's words

Entangled particles exhibit Bell‑inequality violations that are normally interpreted as evidence for non‑local influences or many‑worlds branching. A concrete alternative is proposed: the p‑adic ancestry model, in which the ultrametric structure of a p‑adic tree encodes a fixed hidden‑variable genealogy that determines present‑day correlations. The model replaces co‑existing worlds with a single deterministic ancestor node shared by the particles.

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

Numbers it reports

1. Using the prime $p=3$ (the only value that yields a phase transition in the $p$‑adic Potts model [2]) and a tree depth $L=3$, the model reproduces a CHSH value $S=2.724$, which is approximately 3.7% below the Tsirelson bound $2\sqrt2\approx2.828$.

   from Abstract

2. Numerical evaluation showing that depth $L=3$ yields $S=2.724$, reproducing the Tsirelson bound within $2\%$ (Section 5).

   from 1. Introduction

3. Thus at depth $L=3$ the model yields $S=2.724$, which exceeds the classical bound $2$ and approaches the Tsirelson bound $2.828$ within approximately $3.7\%$.

   from 4.3 CHSH Value as a Function of Depth

4. For the minimal depth that yields a violation beyond the classical bound by less than $5\%$, we find $L=3$ with $S=2.724$.

   from 5. Results

How it is done

1. Correction (version 1.0.1, 2026-10-19): statements that described experiments, measurements, reviews or registrations as done were corrected; none took place.

   from p‑Adic Ancestry Model: A Deterministic Ultrametric Ontology for Quantum Correlations

2. We construct an explicit hidden‑variable scheme on a binary Cayley tree, derive its correlation function, and compute the resulting CHSH parameter.

   from Abstract

3. Ostrowski’s theorem classifies completions of $\mathbb{Q}$ as $\mathbb{R}$ and $\mathbb{Q}_p$ for each prime $p$, and the QNFO framework maps $\mathbb{Q}_3$ to a discrete internal degree of freedom relevant for Bell‑type experiments [8].

   from 1. Introduction

4. Ostrowski’s theorem and quantum numbers.

   from 2. Background and Related Work

What would falsify it

1. The model predicts a specific functional dependence $S(L)=2\sqrt2\,(1-3^{-L})$.

   from 6.2 Falsification Criterion

2. An experiment that measures the CHSH parameter for entangled photon pairs at the standard settings and finds a value inconsistent with any integer $L$ (i.e., not of the form $2\sqrt2\,(1-3^{-L})$ within experimental uncertainty) would falsify the model.

   from 6.2 Falsification Criterion

3. Comprehensive discussion of limitations, falsification criteria, and a concrete experimental protocol (Section 6).

   from 1. Introduction

Where the claim stops

1. Assumption Sensitivity Prime $p=3$ (phase‑transition uniqueness) Changing $p$ to any other prime eliminates the non‑trivial correlation factor $1-p^{-L}$, reducing $S$ below the classical bound.

   from 6.1 Limitations

2. Infinite tree approximation Finite‑size effects (truncation at depth $L_{\max}$) introduce boundary corrections not captured in (4).

   from 6.1 Limitations

3. Ideal detectors and noiseless outcomes Real experiments involve imperfect detection; the model assumes perfect outcomes.

   from 6.1 Limitations

Where to read

1. Abstract167 words · 1 min
2. 1. Introduction351 words · 2 min
3. 2. Background and Related Work306 words · 1 min
4. 3. Methods263 words · 1 min
5. 4. Analysis180 words · 1 min
6. 5. Results107 words · 1 min
7. 6. Discussion227 words · 1 min
8. 7. Practical Implications and Engineering Pathways283 words · 1 min
9. 8. Conclusion147 words · 1 min
10. Correction Notice226 words · 1 min
11. Provenance and AI Involvement229 words · 1 min
12. Appendix A. Divergence report108 words · 1 min