A Hidden Tree Model for Quantum Outcomes

In the paper's words

Quantum superposition underpins quantum computing, interferometry, and macroscopic‑superposition experiments, yet its ontological status remains contested, creating a design bottleneck for error‑correction and measurement hardware. Existing models lack a concrete hidden‑variable construction that reproduces the Hilbert‑space formalism while clarifying why a single outcome is observed. We formulate a deterministic binary‑branching tree of depth d that represents a single hidden‑variable trajectory and define a coarse‑graining map that projects leaf sets onto an effective wavefunction.

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

Numbers it reports

1. Assume a classical simulation of the hidden‑variable tree on a workstation incurs a cost $c_{\text{leaf}} = 10^{-3}$ (arbitrary units) per leaf lookup.

   from (e) Projected cost and performance (projection)

2. The projected simulation cost is $$ C_{\text{sim}} = c_{\text{leaf}}\times L = 10^{-3}\times1024 \approx 1.024. $$ Assumptions: $c_{\text{leaf}}$ approximates the memory‑access cost for a single‑precision operation; the simulation stores only leaf indices and a phase value.

   from (e) Projected cost and performance (projection)

How it is done

1. We also prove three negative results: (i) a power‑law leaf measure normalises only for exponent 1, (ii) non‑negative leaf measures cannot generate interference cross‑terms without encoding phase, and (iii) deterministic leaf assignments satisfy the CHSH bound $|S|\le2$, implying any empirically adequate tree must be non‑local or contextual.

   from Abstract

2. Proof that a power‑law leaf measure $p_{i}=f_{i}^{\gamma}$ normalises only for $\gamma=1$ (Section 4).

   from 1. Introduction

3. A rooted binary tree $\mathcal{T}$ of depth $d\in\mathbb{N}$ is defined such that each node at level $\ell$ ($0\le\ell\le d$) has exactly two children at level $\ell+1$.

   from 3.1 Tree construction

4. The CHSH combination is $$ S = \langle A_{0}B_{0}\rangle + \langle A_{0}B_{1}\rangle + \langle A_{1}B_{0}\rangle - \langle A_{1}B_{1}\rangle. $$ Because each leaf determines a single outcome for each setting, the algebraic maximum of $|S|$ is $2$, reproducing the classical CHSH bound.

   from 4.5 CHSH bound for deterministic leaf assignments

What would falsify it

1. A single observation that would falsify the core claim is the detection of a probability that cannot be expressed as $n/2^{d}$ for any integers $n$ and $d$.

   from 6.2 Failure modes and falsification

2. For example, measuring $P=1/3$ in a two‑path interferometer would contradict the branch‑counting prediction because $1/3$ cannot be written with denominator $2^{d}$.

   from 6.2 Failure modes and falsification

3. More decisive falsification requires observing interference visibility or Bell‑type correlations that exceed the limits derived in Sections 4.4 and 4.5.

   from 6.2 Failure modes and falsification

Where the claim stops

1. Phases $\theta_{k}$ are assumed uniformly distributed; non‑uniform distributions could generate non‑zero cross‑terms but are not derived from the model.

   from 6.1 Limitations

2. The analysis also establishes three negative results: (i) the linear leaf‑fraction measure is forced, (ii) non‑negative leaf measures cannot generate interference cross‑terms without embedding phase, and (iii) deterministic leaf assignments satisfy the classical CHSH bound, implying any empirically adequate tree must be non‑local or contextual.

   from 8. Conclusion

3. Future work will explore phase‑encoding extensions, irregular branching, and higher‑dimensional generalisations to test the robustness of the projection‑artifact interpretation.

   from 8. Conclusion

Where to read

1. Abstract199 words · 1 min
2. 1. Introduction214 words · 1 min
3. 2. Background and Related Work257 words · 1 min
4. 3. Methods166 words · 1 min
5. 4. Analysis308 words · 1 min
6. 5. Results109 words · 1 min
7. 6. Discussion246 words · 1 min
8. 7. Practical Implications and Engineering Pathways525 words · 2 min
9. 8. Conclusion154 words · 1 min
10. Version History22 words · 1 min
11. Provenance and AI Involvement225 words · 1 min