Adelic Ruler‑Independence and Rationality of Standard Model Parameters

In the paper's words

The Standard Model (SM) contains dozens of dimensionless parameters whose measured values appear finely tuned, yet the theory provides no internal mechanism to select them. A gap exists in linking these parameters to a universal mathematical principle. We adopt the hypothesis that physical law must be expressible consistently over every completion of the rational field Q—the real place and all p‑adic places—so that an adelic product formula constrains each parameter.

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

Numbers it reports

1. The experiment would measure the double‑Higgs production cross‑section; a deviation larger than the projected experimental uncertainty (≈ 5 %) would falsify the adelic rationality claim.

   from (c) Smallest next experiment

2. Assuming the measurement uncertainty can be reduced to 2 % (a projection based on current collider design studies), the test would be sensitive to deviations of $\lambda$ from 1 by as little as 0.02 in absolute terms.

   from (e) Projected performance bounds

3. Compute each factor: Infinite place: $$ |\lambda|_{\infty} = \frac{2}{3}\approx0.6666667. $$ $p=2$: $$ \nu_{2}(2)=1,\;\nu_{2}(3)=0\;\Rightarrow\; |\lambda|_{2}=2^{-\nu_{2}(2)+\nu_{2}(3)} = 2^{-1}= \frac{1}{2}=0.5. $$ $p=3$: $$ \nu_{3}(2)=0,\;\nu_{3}(3)=1\;\Rightarrow\; |\lambda|_{3}=3^{-\nu_{3}(2)+\nu_{3}(3)} = 3^{1}=3. $$ All other primes $p\neq2,3$: $$ \nu_{p}(2)=\nu_{p}(3)=0\;\Rightarrow\;|\lambda|_{p}=p^{0}=1. $$ Form the product: $$ \prod_{v}|\lambda|_{v} = |\lambda|_{\infty}\times|\lambda|_{2}\times|\lambda|_{3}\times\prod_{p\neq2,3}1 = \frac{2}{3}\times\frac{1}{2}\times3 = 1. $$ Thus the numerical computation confirms (1) for the specific rational $\lambda=2/3$.

   from 4.2 Concrete numerical example

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 Adelic Ruler‑Independence and Rationality of Standard Model Parameters

2. By formulating a toy scalar coupling $\lambda$ as a rational number $a/b$ and imposing the Ostrowski product identity $\prod_{v}|\lambda|_{v}=1$, we derive that the only positive rational solution satisfying the adelic consistency condition is $\lambda=1$.

   from Abstract

3. Baker‑Rumely and Favre‑Rivera‑Letelier proved an arithmetic equidistribution theorem for points of small height on the Berkovich compactification of the projective line with respect to an adelic measure [1].

   from 2. Background and Related Work

4. For each place $v\in\{\infty\}\cup\{p\text{ prime}\}$ we define the local absolute value $$ |\lambda|_{v}= \begin{cases} \displaystyle\frac{|a|}{|b|}, & v=\infty,\\[6pt] p^{-\nu_{p}(a)+\nu_{p}(b)}, & v=p, \end{cases} $$ where $\nu_{p}(x)$ is the $p$‑adic valuation of $x$ (the exponent of $p$ in its prime factorisation).

   from 3.2 Adelically consistent functional

What would falsify it

1. Assumption Sensitivity Ostrowski product formula holds for the coupling Critical: without it the derivation collapses Coupling is positive and real Moderate: sign change would lead to $\lambda=-1$ One‑loop perturbation theory suffices Low: higher‑loop effects could introduce irrational terms Toy scalar sector captures essential SM features High: SM gauge couplings involve non‑abelian dynamics not modelled here If high‑precision measurements of a SM coupling (e.g., the Higgs self‑coupling) reveal a value significantly different from 1 beyond theoretical uncertainties, the adelic rationality hypothesis would be falsified.

   from 6.1 Limitations

2. High‑energy experiments measure gauge couplings, fermion masses, and the Higgs vacuum expectation value of the Standard Model (SM) to many significant figures, yet the SM offers no internal principle that predicts these numbers.

   from 1. Introduction

3. Historically, attempts to explain these numbers have invoked grand unification, supersymmetry, or anthropic selection, yet each approach introduces additional structure without delivering a unique prediction for the observed values.

   from 2. Background and Related Work

Where the claim stops

1. Assumption Sensitivity Ostrowski product formula holds for the coupling Critical: without it the derivation collapses Coupling is positive and real Moderate: sign change would lead to $\lambda=-1$ One‑loop perturbation theory suffices Low: higher‑loop effects could introduce irrational terms Toy scalar sector captures essential SM features High: SM gauge couplings involve non‑abelian dynamics not modelled here If high‑precision measurements of a SM coupling (e.g., the Higgs self‑coupling) reveal a value significantly different from 1 beyond theoretical uncertainties, the adelic rationality hypothesis would be falsified.

   from 6.1 Limitations

2. While the present work is limited to a simplified scalar sector and relies on several assumptions, it illustrates a concrete pathway by which adelic mathematics might inform fundamental physics.

   from 8. Conclusion

3. Future work should extend the framework to gauge couplings, incorporate higher‑order quantum corrections, and confront the predictions with high‑precision experimental data.

   from 8. Conclusion

Where to read

1. Abstract177 words · 1 min
2. 1. Introduction258 words · 1 min
3. 2. Background and Related Work787 words · 3 min
4. 3. Methods162 words · 1 min
5. 4. Analysis261 words · 1 min
6. 5. Results99 words · 1 min
7. 6. Discussion193 words · 1 min
8. 7. Practical Implications and Engineering Pathways247 words · 1 min
9. 8. Conclusion106 words · 1 min
10. Correction Notice101 words · 1 min
11. Provenance and AI Involvement208 words · 1 min
12. Appendix A. Divergence report125 words · 1 min