One Table, Two Regimes: Standard-Model Particles and Condensed-Matter Excitations as Patterns on the Bruhat-Tits Tree

In the paper's words

Physics maintains two particle catalogs that never meet in one framework: the Standard Model's finite table of elementary fermions and bosons, and the open-ended zoo of quasiparticles and anyons discovered in condensed matter. This paper assembles both catalogs into a single pattern table on the Bruhat-Tits tree. Statistics is read as a tree-automorphism phase; the framework's new content is the table, the regime dictionary, and falsifiability conditions spanning both catalogs.

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

Numbers it reports

1. A phonon is a quantized lattice vibration; a magnon a quantized spin wave; an exciton a bound electron-hole pair; a Cooper pair the bound state underlying superconductivity [2,17,33]; a composite fermion the attached-flux object of the fractional quantum Hall effect [3,20,35].

   from 2.2 The condensed-matter catalog

How it is done

1. The Standard Model lists twelve fermions — six quarks (up, down, charm, strange, top, bottom) and six leptons (electron, muon, tau, and their neutrinos) — all of spin 1/2, plus the gauge bosons (photon, eight gluons, W and Z) of spin 1, and the Higgs scalar of spin 0 [1,2].

   from 2.1 The Standard Model catalog

2. A phonon is a quantized lattice vibration; a magnon a quantized spin wave; an exciton a bound electron-hole pair; a Cooper pair the bound state underlying superconductivity [2,17,33]; a composite fermion the attached-flux object of the fractional quantum Hall effect [3,20,35].

   from 2.2 The condensed-matter catalog

3. Within this research program, the tree's spectral dynamics and transport properties have been developed as physical subjects in their own right [47,48], on foundations that include adelic quantum field theory [54] and the convergence of Ostrowski's theorem with adelic quantum mechanics [55].

   from 3. The Bruhat-Tits Tree as the Common Substrate

4. In 2+1 dimensions the spin-statistics theorem continues to hold for particles obeying braid statistics: the phase is still $e^{2\pi i s}$, with fractional $s$ permitted [4].

   from 4.1 The exchange-phase invariant

What would falsify it

1. A fundamental particle whose statistics phase is not a tree-automorphism phase — for example a fundamental fermion with integer spin, or statistics outside the root-of-unity set — falsifies the statistics reading.

   from 9. Falsifiability Conditions

2. A weakly interacting composite quasiparticle whose effective Compton count is not a rational function of the background counts falsifies the quasiparticle reading.

   from 9. Falsifiability Conditions

3. A physical system exhibiting braid statistics with a phase that is not a root of unity falsifies the anyon reading.

   from 9. Falsifiability Conditions

Where the claim stops

1. The claims are bounded: the premises end at five named inputs, the known entries are retrodictions, the non-abelian regime is open, and the evidential value of the scheme rests on excitations not yet characterized.

   from 11. Conclusion

2. For every known particle the table recapitulates established results; its evidential value lies in the pre-registered address scheme, to be tested against excitations not yet characterized.

   from Abstract

3. The condensed-matter catalog is open, emergent, and approximate.

   from 2.3 The asymmetry

Where to read

1. Abstract251 words · 1 min
2. 1. Introduction404 words · 2 min
3. 2. The Two Catalogs326 words · 1 min
4. 3. The Bruhat-Tits Tree as the Common Substrate274 words · 1 min
5. 4. Statistics as a Tree-Automorphism Phase369 words · 2 min
6. 5. The Unified Pattern Table633 words · 3 min
7. 6. The Regime Dictionary109 words · 1 min
8. 7. Quasiparticles as Rational-Function Nodes163 words · 1 min
9. 8. Epistemic Status236 words · 1 min
10. 9. Falsifiability Conditions152 words · 1 min
11. 10. What a Practitioner Can Do With This392 words · 2 min
12. 11. Conclusion129 words · 1 min