The Adelic Distinction: Physics as Automorphic Representation Theory on the Idele Class Group

In the paper's words

Proposes that the adele ring and idele class group provide the single mathematical object whose automorphic representations generate all known particle physics, gauge interactions, and cosmological parameters. Covers gauge symmetry, spin, charge quantization, three generations, QFT as automorphic harmonic analysis, CPT symmetry, Higgs mechanism, dark matter, dark energy, Riemann hypothesis, quantum gravity, and experimental predictions.

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

Numbers it reports

1. The framework resolves the mutual isolation of number theory, quantum foundations, and cosmology that has persisted for 66–110 years across five independent disciplines.

   from Abstract

How it is done

1. Ostrowski's theorem (1916) establishes that every completion of the rational numbers $\mathbb{Q}$ is either the real Archimedean place $\mathbb{R}$ or a $p$-adic non-Archimedean place $\mathbb{Q}_p$.

   from Abstract

2. Ostrowski's theorem [1] shows that this choice is not forced by logic.

   from 1.1 The Archimedean Assumption

3. The approach is exploratory: it maps the landscape, identifies correspondences, and proposes falsifiable tests.

   from 1.2 The Distinction Network

4. The functional equation of the adelic $L$-function is precisely the mathematical statement of CPT invariance: the theorem that CPT must be conserved follows from the product formula. [speculative]

   from 5.1 CPT as the Product Formula

What would falsify it

1. Experimental signatures — prime-number gravitational wave echoes, parity-violating CMB spectra, and collider fractal substructure — are proposed as falsifiable predictions.

   from Abstract

2. The approach is exploratory: it maps the landscape, identifies correspondences, and proposes falsifiable tests.

   from 1.2 The Distinction Network

3. The $\theta$ angle is an adelic phase, and the product formula forces the total phase to be an integer multiple of $2\pi$, meaning the observable $\theta$ (the Archimedean projection) is zero in the vacuum.

   from 6.3 The Strong CP Problem

Where the claim stops

1. The adelic perspective does not add new assumptions to physics; it removes one — the unjustified restriction to the real numbers — and allows the mathematics of the adeles to determine what remains.

   from 10. Conclusion

2. The standard formalism of physics rests on an unstated assumption: that the real numbers $\mathbb{R}$ are the correct number system for describing physical quantities.

   from 1.1 The Archimedean Assumption

3. The real numbers enter physics through the back door of continuity — the assumption that spacetime is a smooth manifold, that symmetry groups are Lie groups, and that functional integrals run over real-valued fields.

   from 1.1 The Archimedean Assumption

Where to read

1. Abstract273 words · 1 min
2. 1. Introduction426 words · 2 min
3. 2. The Adelic Circle331 words · 1 min
4. 3. Particles as Automorphic Representations374 words · 2 min
5. 4. Quantum Field Theory as Automorphic Harmonic Analysis280 words · 1 min
6. 5. A Unified View of Core Symmetries233 words · 1 min
7. 6. Dark Matter, Neutrinos, and the Strong CP Problem282 words · 1 min
8. 7. Quantum Gravity and Cosmology416 words · 2 min
9. 8. Experimental Signatures208 words · 1 min
10. 9. The Moduli Space and the Constants of Nature192 words · 1 min
11. 10. Conclusion153 words · 1 min
12. Declarations117 words · 1 min