A Deterministic Tree Map That Acts Like Quantum Randomness

In the paper's words

Quantum‑mechanical randomness underlies cryptographic key generation, Monte Carlo simulation, and error‑corrected quantum computation. Existing explanations treat the Born rule as a fundamental stochastic postulate, leaving the origin of randomness opaque. A specific gap is the absence of a deterministic geometric mechanism that reproduces the observed statistical distribution of measurement outcomes.

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

Numbers it reports

1. Compute the leaf weight: $$ w = \frac{1}{p^{k}} = \frac{1}{27}\approx0.037037. $$ Compute the entropy: $$ H_{\text{leaf}} = \log_{2}(27) = \frac{\ln 27}{\ln 2}. $$ Step‑by‑step: $$ \ln 27 = \ln(3^{3}) = 3\ln 3 \approx 3\times1.09861229 = 3.29583687. $$ $$ \ln 2 \approx 0.69314718. $$ $$ \frac{3.29583687}{0.69314718} \approx 4.7548875. $$ Thus $$ H_{\text{leaf}} \approx 4.7548875\ \text{bits}. $$

   from 4.1. Numerical Entropy for $p=3$, $k=3$

2. The normalization factor is the mean density: the $27$ intervals of length $1/27$ carry total weight $1$, so the average height of the discrete density is $1/(1/27)\cdot(1/27)=1$ per unit interval.

   from 4.3. Log‑Periodic Term Amplitude

3. The entropy of the uniform leaf distribution for a depth‑$3$ $3$‑adic tree is $H=4.7548875$ bits, identical to the Shannon entropy of a uniform real interval partitioned into $27$ equal parts.

   from 5. Results

4. The entropy test alone is weak, because any device that outputs each of the $27$ symbols equally often reproduces $H=4.7548875$ bits regardless of mechanism; the discriminating prediction is therefore the log‑periodic component R3, whose absence at the stated threshold would refute the specific projection model rather than determinism in general.

   from 6.2. Falsifiability

5. One would measure the output distribution over $10^{6}$ trials and compute the empirical Shannon entropy.

   from 7.3. Minimal Next Experiment

6. The test would use a firmware implementation of $\varphi$ over the digit registers $(a_{0},a_{1},a_{2})$, driven by a counter that cycles through all $27$ triples; the empirical entropy would be computed as $\hat{H}=-\sum_{i}\hat{P}_{i}\log_{2}\hat{P}_{i}$ with $\hat{P}_{i}=n_{i}/10^{6}$, and the empirical occupancy of each of the $27$ bins would be compared with the expected $10^{6}/27\approx37037$ counts per bin.

   from 7.3. Minimal Next Experiment

How it is done

1. Using the Shannon‑entropy definition from quantum‑random‑number‑generator literature [1] and the phase‑transition condition $p=3$ for the $p$‑adic Potts model [2], we compute the entropy of the projected uniform leaf distribution for $p=3$ and depth $k=3$.

   from Abstract

2. Formal definition of the Monna map $\varphi$ for a $p$‑adic tree and proof of its measure‑preserving property (Section 3).

   from 1. Introduction

3. Quantum‑hardware engineers and cryptographic system designers could employ the deterministic projection model to replace stochastic QRNG modules with deterministic $p$‑adic tree processors that reproduce the same statistical distribution, though the resulting numbers remain algorithmically predictable as a pseudo‑random analogue per [1].

   from 7.1. Potential Users and Design Changes

4. A prototype could enumerate the $27$ leaf states of a depth‑$3$ $3$‑adic tree and implement the Monna map in firmware.

   from 7.3. Minimal Next Experiment

What would falsify it

1. A single observation that would falsify the main claim is a statistically significant deviation from the predicted uniform distribution or entropy value in a controlled hardware test of the Monna‑projection device.

   from 6.2. Falsifiability

2. The entropy test alone is weak, because any device that outputs each of the $27$ symbols equally often reproduces $H=4.7548875$ bits regardless of mechanism; the discriminating prediction is therefore the log‑periodic component R3, whose absence at the stated threshold would refute the specific projection model rather than determinism in general.

   from 6.2. Falsifiability

3. Presentation of a log‑periodic modulation term $\rho(x)=1+\epsilon\cos\!\bigl(2\pi\log_{3}x\bigr)$ with amplitude $\epsilon\approx0.03704$ arising from finite‑depth truncation; the term is discussed as a theoretical prediction without asserted observability.

   from 1. Introduction

Where the claim stops

1. The analysis assumes a finite depth $k$; infinite‑depth limits recover exact uniformity but are unattainable in physical hardware.

   from 6.1. Limitations

2. The model treats measurement outcomes as deterministic projections; it does not address dynamical decoherence mechanisms beyond the static geometry.

   from 6.1. Limitations

3. Quantum random‑number generators (QRNGs) are commercially available, and surveys of the field identify entropy as the central figure of merit for their output quality [1], while the underlying randomness is assumed to be fundamentally indeterminate.

   from 1. Introduction

Where to read

1. Abstract189 words · 1 min
2. 1. Introduction272 words · 1 min
3. 2. Background and Related Work373 words · 2 min
4. 3. Methods253 words · 1 min
5. 4. Analysis316 words · 1 min
6. 5. Results153 words · 1 min
7. 6. Discussion198 words · 1 min
8. 7. Practical Implications and Engineering Pathways350 words · 2 min
9. 8. Conclusion82 words · 1 min
10. Version History19 words · 1 min
11. Provenance and AI Involvement231 words · 1 min