Ostrowski's Classification as a Selection Rule: A Falsifiable Program for the Adelic Shadow of the Standard Model Gauge Group

In the paper's words

Ostrowski's theorem classifies the completions of the rational numbers Q: exactly one Archimedean completion, R, and one p-adic completion Q_p for each prime. Theoretical physics uses only the first. A conjecture in the adelic physics program holds that the observed gauge group SU(3)_c × SU(2)_L × U(1)_Y is the Archimedean shadow of a symmetry structure defined over all places of Q, and that coupling constants and particle representations must satisfy adelic consistency conditions.

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

Numbers it reports

1. We verify the product formula $\prod_v |q|_v = 1$ explicitly ($q = 60$: $60 \cdot 2^{-2} \cdot 3^{-1} \cdot 5^{-1} = 1$), independently reproduce the tabulated QED one-loop coefficient $\beta_0 = 2/\pi = 0.636620$, derive that the global gauge-group ambiguity an adelic selection rule must resolve is finite of order $6$, and compute a quantitative visibility bound for the proposed collider prime-clustering signature: at the documented $10\%$ jet systematic floor, a null result excludes a Möbius-weighted amplitude coefficient $a \geq 0.20$ at harmonic $n = 2$.

   from Abstract

2. A quantitative visibility threshold for the proposed collider prime-clustering signature: at the $10\%$ jet systematic uncertainty floor documented for hadron-collider QCD analyses [1], a null result excludes a Möbius-weighted amplitude coefficient $a \geq 0.20$ at harmonic $n = 2$ (Section 4).

   from 1. Introduction

3. Two points from this work carry directly into the present program: the $10\%$ systematic floor is the visibility threshold any new jet substructure must beat, and the lesson that algorithmic choices (seeds, merge parameters $f_{\mathrm{merge}}$ and $p_{T,\min}$) silently shape rates is a warning for how an adelic fractal signature could be hidden or manufactured by analysis choices [1].

   from 2. Background and Related Work

4. Its master table reports an adelic product formula error below $10^{-12}$, an adelic beta constraint error $\max\|\beta_\infty + \sum \beta_p\|$ below $10^{-13}$, a Veneziano beta at the symmetric point $\beta_\infty(0.5) = -5.372183$, a QED one-loop coefficient $\beta_0 = 2/\pi = 0.636620$, and a compactification ratio $R(0.5) = 8.438606$ [9].

   from 2. Background and Related Work

5. Four computations are performed: the adelic product formula at $q = 60$, the reproduction of the tabulated QED one-loop coefficient, the order of the gauge-group ambiguity, and the collider visibility threshold, followed by the echo-delay spectrum and a worked Schur-multiplier bound.

   from 4. Analysis

6. Systematic uncertainty floor for jet observables: $\sigma_{\mathrm{sys}} = 0.10$ (the report documents jet systematics at the $10\%$ to $50\%$ level with the goal of pushing below $10\%$, and states that precision much better than $10\%$ is not yet guaranteed) [1].

   from 4.4 Collider visibility threshold for the prime-clustering signature

How it is done

1. Ostrowski's theorem classifies the completions of the rational numbers $\mathbb{Q}$: exactly one Archimedean completion, $\mathbb{R}$, and one $p$-adic completion $\mathbb{Q}_p$ for each prime.

   from Abstract

2. Ostrowski's theorem shows this choice is not forced: the rational numbers $\mathbb{Q}$ admit exactly one Archimedean completion and one $p$-adic completion for each prime, and these are all the completions [8], [9], [10].

   from 1. Introduction

3. The Tevatron Run 2 experiments each accumulated over $1\;\mathrm{fb}^{-1}$ of high-transverse-momentum data, enabling comparisons between theory and experiment at the few-percent level, and the report of the QCD working group documents that jet systematic uncertainties stood at the $10\%$ to $50\%$ level with the explicit goal of pushing below $10\%$ [1].

   from 2. Background and Related Work

4. The conjecture under test is stated in three parts, each with a falsification condition.

   from 3. Methods

What would falsify it

1. The gap this paper addresses is that the conjecture has never been stated with explicit falsification criteria tied to existing experimental precision budgets.

   from Abstract

2. We verify the product formula $\prod_v |q|_v = 1$ explicitly ($q = 60$: $60 \cdot 2^{-2} \cdot 3^{-1} \cdot 5^{-1} = 1$), independently reproduce the tabulated QED one-loop coefficient $\beta_0 = 2/\pi = 0.636620$, derive that the global gauge-group ambiguity an adelic selection rule must resolve is finite of order $6$, and compute a quantitative visibility bound for the proposed collider prime-clustering signature: at the documented $10\%$ jet systematic floor, a null result excludes a Möbius-weighted amplitude coefficient $a \geq 0.20$ at harmonic $n = 2$.

   from Abstract

3. What changes if the conjecture fails is equally valuable: part of the failure would be an internal, algebraic falsification requiring no new experiment, establishing that Ostrowski's classification constrains arithmetic but not physics.

   from 1. Introduction

Where the claim stops

1. What the results do and do not establish.

   from 6. Discussion

2. R1 and R2 establish that the adelic program's arithmetic core is internally consistent; they do not establish that physics is adelic.

   from 6. Discussion

3. Second, a coupling failure: measured couplings, propagated to local components, violate $\prod_v |g_v|_v = 1$ beyond combined uncertainties, falsifying Test T2 directly.

   from 6. Discussion

Where to read

1. Abstract231 words · 1 min
2. 1. Introduction550 words · 2 min
3. 2. Background and Related Work1,379 words · 6 min
4. 3. Methods427 words · 2 min
5. 4. Analysis1,334 words · 6 min
6. 5. Results329 words · 1 min
7. 6. Discussion794 words · 3 min
8. 7. Practical Implications and Engineering Pathways798 words · 3 min
9. 8. Conclusion209 words · 1 min
10. Provenance and AI Involvement225 words · 1 min