From Peaks to Pixels

In the paper's words

This paper demonstrates the structural isomorphism between the physical quantization of continuous waves (e.g., peak counting) and the information-theoretic process of signal digitization. Using a computational simulation of 50 randomized waveforms, we compare the Shannon entropy of discrete peak distributions against that of digital samples. Results show a strong statistical correlation (r ≈ 0.77) and minimal entropy difference (~0.04 bits) between the two methods, suggesting that “discrete math” in physics is informationally equivalent to digital sampling governed by the Nyquist-Shannon theorem.

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

Numbers it reports

1. Results show a strong statistical correlation ($r \approx 0.77$) and minimal entropy difference (~0.04 bits) between the two methods, suggesting that “discrete math” in physics is informationally equivalent to digital sampling governed by the Nyquist-Shannon theorem.

   from Abstract

2. These waveforms were constructed as superpositions of five random sinusoidal components with frequencies ranging from 1 to 20 Hz, overlaid with a Gaussian noise floor to mimic the thermal fluctuations inherent in real physical systems (Adcock, 2016).

   from 4.1 Simulation Data Overview

3. In observing the raw entropy values, we found that both the Peak Quantization (Method A) and the Nyquist Digitization (Method B) produced Shannon entropy scores consistently falling within the narrow range of 3.9 to 4.0 bits per symbol (Garcia, 2015).

   from 4.2 Peak-Entropy Vs

4. For instance, in a representative trial, the peak entropy ($H_{peaks}$) was calculated at 3.98 bits, while the corresponding sample entropy ($H_{samples}$) was 3.97 bits.

   from 4.2 Peak-Entropy Vs

5. The mean absolute difference between the two entropy measures was calculated to be approximately 0.043 bits, representing a deviation of less than 1.5% relative to the total entropy.

   from 4.3 Correlation Analysis

6. We have demonstrated that the Shannon entropy of discrete peaks converges with the Shannon entropy of digital samples with a correlation of $r \approx 0.765$ and a negligible mean difference of 0.043 bits.

   from 4.6 Results Summary

How it is done

1. Using a computational simulation of 50 randomized waveforms, we compare the Shannon entropy of discrete peak distributions against that of digital samples.

   from Abstract

2. The fidelity of this translation is governed by the Nyquist-Shannon sampling theorem, which dictates that a continuous signal can be reconstructed with high fidelity from its discrete samples provided the sampling rate exceeds twice the signal’s bandwidth.

   from 1.2 The Signal Processing

3. While this approach abstracts away the specific dynamics of the Schrödinger equation, it isolates the fundamental information-processing core of the quantization problem.

   from 1.4 Scope and Boundaries

4. Energy levels in bound states are quantized, and the detection of a particle occurs at a localized point in space-time, not as a smeared field.

   from 2.1 Wave Mechanics and

What would falsify it

1. However, the observable world—the world of measurement phenomena—manifests as discrete quanta.

   from 2.1 Wave Mechanics and

2. This juxtaposition creates the central tension of quantum mechanics: the substrate is continuous, yet the observable is discrete.

   from 2.1 Wave Mechanics and

3. If there exists a minimum observable length scale, such as the Planck length, then the physical universe effectively has a “cutoff frequency.” According to the logic of the Sampling Theorem, this cutoff implies that the continuous geometry of General Relativity and the discrete structures of quantum gravity are isomorphic representations of the same underlying information structure (Kempf, 2021).

   from 2.5 Spacetime as a

Where the claim stops

1. It is important to acknowledge that this classical proxy cannot capture non-local quantum correlations or entanglement; however, it is sufficient to demonstrate the emergence of discrete entropy from continuous fields.

   from 1.4 Scope and Boundaries

2. We do not claim to resolve the ontological status of the wavefunction itself, but rather to map the structural constraints that apply to any process extracting discrete values from a continuum.

   from 1.4 Scope and Boundaries

3. However, this emergent discreteness is often dismissed as a mere approximation rather than a fundamental structural property.

   from 1.1 The Illusion of

Where to read

1. Demonstrating25 words · 1 min
2. Abstract112 words · 1 min
3. Keywords12 words · 1 min
4. 1.0 Introduction909 words · 4 min
5. 2.01,177 words · 5 min
6. 3.01,191 words · 5 min
7. 4.0 Results: Entropy1,103 words · 5 min
8. 5.0 Discussion: The Epistemic1,206 words · 5 min
9. 6.01,068 words · 5 min
10. Appendices381 words · 2 min