Base-Invariant Number-Theoretic Patterns in Fundamental Constants

In the paper's words

This presents a systematic, first‑principles investigation of the mathematical structures that underlie fundamental constants, freed from the arbitrariness of numeral‑system representation. By focusing exclusively on properties that are independent of any choice of base we uncover striking patterns that suggest a deep number‑theoretic origin for dimensionless constants. We introduce the Principle of Base Invariance—the requirement that fundamental laws be expressible without privileging any base—and develop its consequences through the Monna map and the adelic framework, which treats all completions of ℚ on equal footing (Ostrowski’s theorem).

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

Numbers it reports

1. Yet the fundamental constants of nature—the fine‑structure constant $\alpha \approx 1/137.036$, the proton‑electron mass ratio $\mu \approx 1836.15$, and the dimensionless ratios that appear in quantum field theory and cosmology—are independent of any system of units.

   from 1.

2. The proton‑electron mass ratio $\mu$ also exhibits large quotients ( 34 , 56 ), suggesting similar “near‑rational” behavior.

   from 3.2 Physical Constants

3. Known bounds for selected constants: Constant $\mu(x)$ (current bound) Notes $\pi$ $\mu(\pi) \le 7.103205\ldots$ (Salikhov, 2008) $e$ $\mu(e) = 2$ proven (rational approximations are optimal) $\phi$, $\sqrt{2}$ $\mu = 2$ Roth’s theorem for algebraic numbers $\gamma$ unknown, believed to be 2 $\zeta(3)$ $\mu(\zeta(3)) \le 5.513891\ldots$ (Rhin & Viola, 2001) $\alpha$, $\alpha^{-1}$ unknown no specific bounds published The relatively large partial quotients for $\pi$ and $\alpha^{-1}$ suggest these numbers admit very good rational approximations, consistent with a higher irrationality measure.

   from 4.1 Irrationality Measure

4. For example: $\pi$‑adic scaling: Rotational phenomena are described by fields on $\mathbb{Q}_\pi$, with $\pi$ acting as the fundamental angular‑to‑linear scaling operator. $\phi$‑adic scaling: Growth processes and self‑similar structures are described on $\mathbb{Q}_\phi$, with $\phi$ as the fundamental growth‑ratio operator. $e$‑adic scaling: Exponential decay and entropic processes are described on $\mathbb{Q}_e$, with $e$ as the fundamental continuous‑growth operator. $\alpha^{-1}$‑adic scaling: Electromagnetic interactions are described on $\mathbb{Q}_{\alpha^{-1}}$, with $\alpha^{-1} \approx 137.036$ as the fundamental coupling‑strength operator.

   from 7.1 From Real

5. They also offer new perspectives on old puzzles: why $\alpha \approx 1/137$, why the proton‑electron mass ratio is ~1836, and why the cosmological constant is so small yet non‑zero.

   from 8. Conclusion

How it is done

1. We introduce the Principle of Base Invariance —the requirement that fundamental laws be expressible without privileging any base—and develop its consequences through the Monna map and the adelic framework, which treats all completions of $\mathbb{Q}$ on equal footing (Ostrowski’s theorem).

   from From

2. Section 5 introduces the adelic framework (Ostrowski’s theorem) and generalizes p‑adic analysis to q‑adic scaling operations, where constants like $\pi , \varphi , e$, and $\alpha ^{-1}$ are interpreted as fundamental scaling operators.

   from 1.

3. Measure The irrationality measure $\mu(x)$ (also called the approximation exponent ) is defined as $$ \mu(x) = \sup\left\{ \mu \in \mathbb{R} : \left| x - \frac{p}{q} \right| \lt \frac{1}{q^\mu} \text{ has infinitely many solutions } (p,q) \in \mathbb{Z}^2 \right\}. $$ For rational $x$, $\mu(x)=1$; for algebraic irrationals, $\mu(x)=2$ (Roth’s theorem).

   from 2.2 Irrationality

4. Continued Fraction Analysis of Fundamental Constants We compute continued fraction expansions for a selection of mathematical and physical constants using high‑precision approximations (typically 15–20 decimal digits).

   from 3.

What would falsify it

1. A proof of algebraic independence would show that no finite polynomial relation with rational coefficients links these constants, reinforcing their status as independent fundamental parameters.

   from 4.3 Algebraic

Where the claim stops

1. This document has three primary objectives: Catalog base‑invariant patterns in fundamental mathematical and physical constants, with emphasis on continued fraction expansions and Diophantine approximation properties.

   from 1.

2. Convergents $p_n/q_n$ are the rational approximations obtained by truncating the expansion after $a_n$.

   from 2.1 Continued

3. They satisfy $$ \left| x - \frac{p_n}{q_n} \right| \lt \frac{1}{q_n^2}, $$ and the quality of approximation is controlled by the size of $a_{n+1}$: $$ \left| x - \frac{p_n}{q_n} \right| \approx \frac{1}{a_{n+1} q_n^2}. $$ Thus a large partial quotient signals an exceptionally good rational approximation.

   from 2.1 Continued

Where to read

1. From275 words · 1 min
2. Contents39 words · 1 min
3. 1.378 words · 2 min
4. 2.511 words · 2 min
5. 3.963 words · 4 min
6. 4.650 words · 3 min
7. 5. Adelic, p‑Adic,1,352 words · 6 min
8. 6.709 words · 3 min
9. 7. Base‑Free702 words · 3 min
10. 8. Conclusion264 words · 1 min
11. 9. References349 words · 2 min