Superposition as a Projection Artifact of Discrete Tree Geometry: A Deterministic Toy Model for Quantum Statistics

In the paper's words

Quantum technologies rely on the Born rule, interference, and Bell‑type correlations, yet the ontological status of superposition remains debated. Existing interpretations either postulate a fundamentally probabilistic wavefunction or invoke many‑world branching, leaving the measurement problem unresolved. This work fills the gap by constructing a deterministic binary‑tree model of depth N whose hidden leaf λ is projected onto a coarse‑grained measurement basis.

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

Numbers it reports

1. Derivation of a per‑class error bound $2^{-(N+1)}$ and a worked example for $|\alpha|^{2}=1/3$ at $N=10$ (Section 4.1).

   from 1. Introduction

2. Proof that Bell‑CHSH correlations obey the classical bound $S\le2$ for any local hidden‑variable assignment on the tree, together with explicit strategies attaining $S=2$ (Section 4.4).

   from 1. Introduction

3. A bound on rounding errors for non‑dyadic amplitudes demonstrates that the maximal deviation scales as $2^{-(N+1)}$, providing the theoretical limit used in Section 4.4 [7].

   from 2. Background and Related Work

4. Then $2^{N}x = 1024/3 = 341.333\ldots$, so $$ n_{1}= \left\lfloor 341.333\ldots + 0.5 \right\rfloor = \left\lfloor 341.833\ldots \right\rfloor = 341 . $$ The model probability is $$ \frac{n_{1}}{2^{N}} = \frac{341}{1024}=0.3330078125 . $$ The deviation from the Born value $1/3=0.333333\ldots$ is $$ \left|0.3330078125 - 0.333333\ldots\right| = \frac{1}{3072}\approx 3.2552\times10^{-4}, $$ which satisfies the bound $2^{-(N+1)}=2^{-11}=4.8828\times10^{-4}$.

   from 4.4 Per‑Class Error Bound and Worked Example

5. Error bound verification : For $|\alpha|^{2}=1/3$ at $N=10$, the deviation from the Born value is $3.2552\times10^{-4}$, well within the theoretical bound $2^{-(N+1)}$.

   from 5. Results

6. Total‑variation bound : At $N=20$ the maximal possible deviation from quantum predictions is $9.5367\times10^{-7}$, setting an experimental sensitivity target.

   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 Superposition as a Projection Artifact of Discrete Tree Geometry: A Deterministic Toy Model for Quantum Statistics

2. Derivation of a per‑class error bound $2^{-(N+1)}$ and a worked example for $|\alpha|^{2}=1/3$ at $N=10$ (Section 4.1).

   from 1. Introduction

3. A bound on rounding errors for non‑dyadic amplitudes demonstrates that the maximal deviation scales as $2^{-(N+1)}$, providing the theoretical limit used in Section 4.4 [7].

   from 2. Background and Related Work

4. We define a coarse‑graining map $$ \Gamma:\{\text{leaves}\}\rightarrow\{0,1\}, $$ which partitions the leaf set into two equivalence classes: $$ \mathcal{C}_{0}=\{\ell_{k}\mid \Gamma(\ell_{k})=0\},\qquad \mathcal{C}_{1}=\{\ell_{k}\mid \Gamma(\ell_{k})=1\}. $$ The map is constructed by grouping leaves according to the most significant bit of the binary label $k$: $$ \Gamma(\ell_{k})= \begin{cases} 0 & \text{if } \big\lfloor k/2^{N-1}\big\rfloor =0,\\[4pt] 1 & \text{if } \big\lfloor k/2^{N-1}\big\rfloor =1. \end{cases} $$ Thus each class contains $2^{N-1}$ leaves for a balanced tree.

   from 3.2 Coarse‑Graining Map

What would falsify it

1. Any experiment that resolves deviations larger than this bound falsifies the model.

   from 3.6 Falsifiability Criterion

2. The model makes concrete falsifiable predictions: In a cascade of three 50:50 beam splitters (depth $N=3$) with a detector that groups the first four paths into outcome 0 and the remaining four into outcome 1, the observed frequencies should be $0.625$ vs. $0.375$ if the bias $b=1$ is present.

   from 6.2 Falsifiability

3. Observation of exact $0.5$–$0.5$ would falsify the biased‑tree hypothesis.

   from 6.2 Falsifiability

Where the claim stops

1. Uniform prior assumption : Derivations assume an a priori uniform distribution over leaves.

   from 6.1 Limitations

2. Interference‑like correlations and visibility are reproduced by integer arithmetic on overlapping class volumes, and Bell‑type correlations obey the classical CHSH bound, demonstrating that non‑local tree connections alone cannot generate quantum violations.

   from 8. Conclusion

3. Future work will explore extensions to multi‑qubit systems, connections to Hilbert‑space formalism, and physical mechanisms that could generate coarse‑graining bias.

   from 8. Conclusion

Where to read

1. Abstract185 words · 1 min
2. 1. Introduction266 words · 1 min
3. 2. Background and Related Work261 words · 1 min
4. 3. Methods467 words · 2 min
5. 4. Analysis516 words · 2 min
6. 5. Results171 words · 1 min
7. 6. Discussion291 words · 1 min
8. 7. Practical Implications and Engineering Pathways293 words · 1 min
9. 8. Conclusion119 words · 1 min
10. Correction Notice303 words · 1 min
11. Provenance and AI Involvement236 words · 1 min