COMMUTATIVE PHYSICS

In the paper's words

Mathematics serves as the fundamental language through which the intricate laws and diverse phenomena of the physical universe are articulated, modeled, and predicted. The selection of mathematical tools is not a trivial matter; it profoundly shapes our comprehension of reality. Modern physics predominantly relies on mathematical formalisms that, while undeniably powerful and successful in numerous domains, are increasingly being scrutinized for their potential limitations in fully capturing the intrinsic structure of the universe at its most fundamental levels.

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

Numbers it reports

No sentence of this paper reports a figure with a unit or a quantity word, so none is shown.

How it is done

1. Through the use of case studies and comparative analyses, this report aims to highlight the potential advantages that such a geometric approach might offer in addressing some of the persistent challenges that confront modern physics.

   from 1. Introduction

2. An alternative approach to this challenge lies in the potential of using exact symbolic ratios involving fundamental geometric constants like $\pi$ and $\varphi$.

   from 3.2

3. Similarly, singularities in black hole physics, where quantities like density are predicted to become infinite, often arise from mathematical assumptions involving division by terms that approach zero (1/r as r → 0), rather than from direct observational evidence of such infinities.20 The concept of absolute nothingness represented by zero might not have a direct physical counterpart at fundamental levels, suggesting the need for alternative approaches to describe states of minimal excitation or extreme density.47 The quantum vacuum, far from being empty, exhibits fluctuations and zero-point energy.

   from 5.1 The Challenge of Zero

4. The framework’s approach to zero handling, negatives, and imaginary numbers offers alternative interpretations that might be more physically meaningful.

   from 7. Mathematical

What would falsify it

The paper's text states no explicit falsification criterion; the limits below are what it says about where the claim stops.

Where the claim stops

1. The prevalence of counting using ten fingers has been a primary driver in the dominance of this system.1 However, a comparative analysis of different cultures reveals that alternative number systems have existed and functioned effectively.

   from 2.1 The Base-10 Number System

2. A critical limitation shared by these conventional number systems, including base-10, is their inherent difficulty in providing a finite representation for infinite continua.

   from 2.1 The Base-10 Number System

3. Irrational numbers, such as $\pi$, require an infinite sequence of digits in any integer base for their exact representation, including base-10.9 This necessity for infinite representation leads to truncation in practical calculations, introducing approximation errors.

   from 2.1 The Base-10 Number System

Where to read

1. 1. Introduction533 words · 2 min
2. 2.1,345 words · 6 min
3. 3.608 words · 3 min
4. 4.986 words · 4 min
5. 5.1,261 words · 5 min
6. 6. $\Pi$ and $\Phi$ as Foundational492 words · 2 min
7. 7. Mathematical309 words · 1 min
8. 8. Summary: Toward Geometric1,137 words · 5 min