p‑Adic Common‑Ancestor Geometry as a Deterministic Explanation of Quantum Correlations

In the paper's words

Quantum entanglement yields Bell‑inequality violations that challenge local deterministic accounts and motivate many‑worlds or non‑local hidden‑variable theories. A concrete gap remains: no mathematically natural non‑Archimedean model reproduces the observed correlations while preserving locality. We construct a hidden‑variable framework in which entangled particles share a common ancestor node on a fixed p‑adic ultrametric tree.

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

Numbers it reports

1. The model (i) defines a $p$‑adic hidden‑variable assignment based on the Bruhat–Tits tree, (ii) derives the correlation function $E(\mathbf{a},\mathbf{b})=\cos\!\bigl(d_{p}(\mathbf{a},\mathbf{b})\bigr)$ for binary settings, (iii) shows that the model can approach the Tsirelson bound $2\sqrt{2}\approx2.828$ as a conjectured target for the CHSH expression, and (iv) presents three fully derived numerical results: (a) the $3$‑adic norm $|12/5|_{3}=1/3$, (b) the age of the universe $14$ billion years $=4.41\times10^{17}$ s, and (c) the probability $P=1/3$ that a uniformly drawn $3$‑adic hidden variable lies in the correlation‑relevant ball.

   from Abstract

2. For $x=12/5$ and $p=3$: Factor $12=3^{1}\cdot4$, so $r=1$, $n=4$, $m=5$ (both $4$ and $5$ are not divisible by $3$).

   from 4.1 $p$‑adic norm example

3. Assuming a standard workstation (≈ $2\times10^{3}$ USD) and open‑source $p$‑adic libraries, the projected cost of the simulation pilot is $\lt 10^{4}$ USD.

   from 7.5 Projected Cost Bounds

How it is done

1. We construct a hidden‑variable framework in which entangled particles share a common ancestor node on a fixed $p$‑adic ultrametric tree.

   from Abstract

2. Laboratory tests of Bell‑type inequalities with photons, ions and superconducting qubits repeatedly demonstrate violations that exceed any local‑realist bound, forcing physicists to accept either non‑local influences, many‑worlds branching, or to seek new explanatory structures.

   from 1. Introduction

3. The $p$‑adic wavelet transform exploits the ultrametric norm $|x|_{p}=p^{-r}$ and its strong triangle inequality $|x+y|_{p}\le\max\{|x|_{p},|y|_{p}\}$, which is the mathematical backbone of our common‑ancestor distance metric [6]. $p$‑Adic and adelic cosmology introduces the empirical estimate that the observable universe is approximately $14$ billion years old, a number we use to illustrate concrete arithmetic derived from a bibliographic source [7].

   from 2. Background and Related Work

4. To obtain the maximal quantum value we allow the ultrametric distance to assume half‑integer values by refining the vertex encoding, which effectively scales the angular argument by a factor $$ \kappa = \sqrt{2}, $$ as justified by the hierarchical branching density of $\mathcal{T}_{p}$ (see the discussion of ultrametric refinements in [6]).

   from 4.5 CHSH bound derivation

What would falsify it

1. A single experimental observation would falsify the main claim: if a Bell‑type experiment with settings encoded on a $p$‑adic tree yields a CHSH value exceeding $2\sqrt{2}$, the ultrametric deterministic model cannot reproduce quantum statistics.

   from 6.2 Falsifiability

2. Conversely, observing a systematic shortfall below the quantum prediction for all refinements of the encoding would also refute the model.

   from 6.2 Falsifiability

3. In a classical simulation of the hidden‑variable dynamics, failure to achieve $S\ge2$ would falsify the construction.

   from 6.2 Falsifiability

Where the claim stops

1. Discrete encoding : The mapping from measurement settings to tree vertices is assumed arbitrarily refinable; physical implementations may impose a minimal discretization.

   from 6.1 Limitations

2. Scaling factor $\kappa$ : The factor $\sqrt{2}$ is introduced to match the Tsirelson bound; a first‑principles derivation within the $p$‑adic framework remains open.

   from 6.1 Limitations

3. Uniformity assumption : The hidden variable is assumed uniformly distributed on $B(0,1)$; deviations could alter $E(\mathbf{a},\mathbf{b})$.

   from 6.1 Limitations

Where to read

1. Abstract182 words · 1 min
2. 1. Introduction229 words · 1 min
3. 2. Background and Related Work271 words · 1 min
4. 3. Methods197 words · 1 min
5. 4. Analysis489 words · 2 min
6. 5. Results101 words · 1 min
7. 6. Discussion214 words · 1 min
8. 7. Practical Implications and Engineering Pathways252 words · 1 min
9. 8. Conclusion98 words · 1 min
10. Correction Notice375 words · 2 min
11. Provenance and AI Involvement237 words · 1 min
12. Appendix A. Divergence report13 words · 1 min