A Geometry-Based Origin for Quantum Probabilities

In the paper's words

Quantum theory postulates the Born rule without a deeper structural justification, and Bell‑type correlations are usually interpreted as non‑local. This gap hinders deterministic reconstructions of quantum mechanics and inflates the engineering overhead of quantum‑information technologies that must treat randomness as a primitive. We model the static ultrametric Bruhat–Tits tree of p‑adic balls as a branching sample space and derive quantum probabilities from the Haar (topological) measure of these balls.

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

Numbers it reports

1. This observation provides an alternative derivation of the result of Section 4.1: the recursive splitting of a ball into $p$ subballs yields a wavelet basis in which each subwavelet has norm reduced by a factor $1/p$, and therefore the measure of a depth‑$k$ ball is $p^{-k}$.

   from 1. Wavelet representation of Haar‑measure balls

2. (see [8]) states that quantum error correction and Quantum Darwinism cannot coexist above a logical fidelity $F_{L}\gt 0.874$.

   from 5. Quantised redundancy in an ultrametric setting

3. The cumulative redundancy after depth $k$ is therefore $$ R(k)=\sum_{j=1}^{k}\frac{1}{p^{j}}= \frac{1/p\;(1- (1/p)^{k})}{1-1/p} =1-\frac{1}{p^{k}}. $$ For $p=2$ and $k=3$ we obtain $R(3)=1-2^{-3}=0.875$, which is just above the critical fidelity $0.874$.

   from 5. Quantised redundancy in an ultrametric setting

How it is done

1. Using the $p$‑adic norm $|x|_{p}=p^{-r}$, the definition of closed balls $B_{r}(a)=\{x\in\mathbb{Q}_{p}\mid|x-a|_{p}\le p^{r}\}$, and the identification of the ball hierarchy with the tree, we prove that a ball at depth $k$ carries measure $p^{-k}$.

   from Abstract

2. Construction of a $p$‑adic branching model on the Bruhat–Tits tree and proof that its Haar measure of a ball at depth $k$ equals $p^{-k}$ (Section 4).

   from 1. Introduction

3. Section 2.1 of [1] defines the $p$‑adic norm $|x|_{p}=p^{-r}$ for $x=p^{r}\frac{n}{m}$, which we use to assign a size to each ball.

   from 2. Background and Related Work

4. We model the sample space as the infinite rooted $p$‑adic tree $\Gamma$ of order $p$, where each vertex has $p$ children.

   from 3. Methods

What would falsify it

1. Superdeterministic shared‑ancestor correlation The model relies on pre‑existing correlations; experimental tests that close the superdeterminism loophole would falsify the claim.

   from 7.1 Limitations

2. A single observation that would falsify the main result is an empirical probability $P$ that cannot be expressed as $p^{-r}$ for any integer $r$ and any fixed prime $p$ consistent across all outcomes (e.g., $P=0.4$ in a binary measurement).

   from 7.1 Limitations

3. Current quantum‑technology platforms—quantum computers, quantum‑secure communication links, and high‑precision metrology—rely on the Born rule to predict measurement outcomes.

   from 1. Introduction

Where the claim stops

1. Assumption Sensitivity Static infinite Bruhat–Tits tree Any dynamical modification of the tree would alter the measure distribution.

   from 7.1 Limitations

2. Mapping of amplitudes to integer radii Only amplitudes whose squared moduli are rational powers of a single prime $p$ are exactly reproduced; irrational probabilities appear as approximations.

   from 7.1 Limitations

3. A single observation that would falsify the main result is an empirical probability $P$ that cannot be expressed as $p^{-r}$ for any integer $r$ and any fixed prime $p$ consistent across all outcomes (e.g., $P=0.4$ in a binary measurement).

   from 7.1 Limitations

Where to read

1. Abstract191 words · 1 min
2. 1. Introduction241 words · 1 min
3. 2. Background and Related Work229 words · 1 min
4. 3. Methods191 words · 1 min
5. 4. Analysis288 words · 1 min
6. 5. Results86 words · 1 min
7. 6. Extended Derivations and Worked Examples1,159 words · 5 min
8. 7. Discussion262 words · 1 min
9. 8. Practical Implications and Engineering Pathways292 words · 1 min
10. 9. Conclusion133 words · 1 min
11. Version History19 words · 1 min
12. Provenance and AI Involvement230 words · 1 min