Reformulating Quantum Mechanics on Bruhat–Tits Trees: A Deterministic Boundary‑Measure Framework

In the paper's words

Quantum foundations confront the measurement problem and the postulated Born rule P_i=|c_i|². Existing platforms—type‑I parametric‑down‑conversion sources delivering 10 kHz coincidences at 200 mW pump power [1] and superconducting processors achieving 99.90\% single‑qubit gate fidelity [2]—demonstrate the empirical success of the standard formalism but leave its axioms unexplained. We rebuild finite‑dimensional quantum mechanics on the ultrametric state space defined by the Bruhat–Tits tree T_p of a prime p.

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

Numbers it reports

1. Existing platforms—type‑I parametric‑down‑conversion sources delivering $10\;\text{kHz}$ coincidences at $200\;\text{mW}$ pump power [1] and superconducting processors achieving $99.90\%$ single‑qubit gate fidelity [2]—demonstrate the empirical success of the standard formalism but leave its axioms unexplained.

   from Abstract

2. Tianyan’s quantum cloud platform operates $105$‑qubit superconducting processors with single‑qubit gate fidelity $99.90\%$, two‑qubit gate fidelity $99.56\%$, and readout fidelity $98.7\%$ [2].

   from 2. Background and Related Work

3. Our bare tree‑derived probability $0.125$ lies far below this threshold, indicating that additional entanglement resources are required for advantage.

   from 2. Background and Related Work

4. Deterministic container assignments satisfy the CHSH bound $|S|\le2$, implying that Bell‑inequality violation would falsify the framework (Section 4.5).

   from 5. Results

5. An experiment preparing a qubit in the state $(3/5,4/5)$ and measuring the outcome frequencies over $N=10^{5}$ trials should yield frequencies $f_{1}\approx0.36$, $f_{2}\approx0.64$ with binomial standard deviation $\sigma=\sqrt{P(1-P)/N}\approx0.0015$.

   from 6.2 Falsification criterion

How it is done

1. For a qubit with rational amplitudes $c_1=3/5$, $c_2=4/5$ on a $p=5$ tree with $N=2$ levels we derive container measures $9/25=0.36$ and $16/25=0.64$, exactly matching $|c_i|^2$.

   from Abstract

2. Proof that the uniform boundary measure on $\partial T_{p}$ yields the Born rule for all states whose squared amplitudes are rationals of the form $m_i/p^{N}$ (Section 4).

   from 1. Introduction

3. Ontological models assign deterministic outcomes to hidden variables constrained on a sphere and are limited by the Kochen–Specker theorem, which imposes contextuality constraints [4].

   from 2. Background and Related Work

4. A measurement with outcomes $\{1,\dots,d\}$ corresponds to the deterministic event that the hidden boundary point $x$ lies in container $C_{i}$.

   from 3.6 Measurement as deterministic projection

What would falsify it

1. Observation of $|f_{i}-P_{i}|\gt 3\sigma$ for either outcome would falsify the boundary‑measure hypothesis.

   from 6.2 Falsification criterion

2. Identification of experimental signatures that would falsify the boundary‑measure hypothesis (Section 6).

   from 1. Introduction

3. Bohmian mechanics predicts opposite‑sign time correlations for a harmonic‑oscillator ground state compared with standard quantum mechanics, highlighting the sensitivity of observable predictions to the underlying representation [7].

   from 2. Background and Related Work

Where the claim stops

1. Uniform boundary measure assumes perfect symmetry; any physical bias (e.g., asymmetric coupling) would break the correspondence.

   from 6.1 Limitations

2. Deterministic container assignments obey the classical CHSH bound, so the framework cannot reproduce observed Bell‑inequality violations without extension.

   from 6.1 Limitations

3. Generalizing quantum mechanics for quantum spacetime argues that the textbook formulation assumes a fixed background and must be extended to fluctuating geometry [5].

   from 2. Background and Related Work

Where to read

1. Abstract200 words · 1 min
2. 1. Introduction291 words · 1 min
3. 2. Background and Related Work461 words · 2 min
4. 3. Methods336 words · 1 min
5. 4. Analysis325 words · 1 min
6. 5. Results120 words · 1 min
7. 6. Discussion154 words · 1 min
8. 7. Practical Implications and Engineering Pathways235 words · 1 min
9. 8. Conclusion119 words · 1 min
10. Provenance and AI Involvement225 words · 1 min
11. Appendix A. Divergence report166 words · 1 min
12. Appendix B. Claim attribution40 words · 1 min