Why Tiny Symmetry Models Cannot Build the Standard Model

In the paper's words

The Standard Model (SM) requires a continuous gauge group SU(3)× SU(2)× U(1) with infinitely many charge sectors, while finite‑group Walker‑Wang constructions provide only finitely many superselection sectors. Existing proposals therefore face a gap: no demonstration that a finite‑group braided fusion category can yield the SM gauge symmetry after coarse‑graining. We combine (i) the classification of finite‑group 2‑gauge theories [2], (ii) the excess‑zero property of finite Coxeter groups [1], (iii) the density‑of‑states formula for finite‑group symmetry [3], and (iv) the factorisation of gauge symmetry on intervals [4] to prove a no‑go theorem.

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

Numbers it reports

1. Assuming the standard result that $Z(\mathrm{Rep}(G))$ contains exactly $|G|$ simple objects, we obtain a concrete bound for the candidate group $G=SL(2,3)$: $|G|=24$ and $Z(\mathrm{Rep}(SL(2,3)))$ has only five simple objects.

   from Abstract

2. Using (2) with $M=24$ gives a provisional bound $N_{\text{charges}}\le 24$.

   from 4.3. Numerical projection for $G=SL(2,3)$

3. Theorem 1 (Section 4.1) using bound (1).

   from 5. Results

4. For $k=2$, $$ M_{2}=3^{2}\bigl(9^{2}-1\bigr)=9\,(81-1)=9\times80=720. $$ Thus the bound on charge sectors would be $N_{2}=720$.

   from 1. Scaling of the charge bound with group order

5. The total number of distinct Lorentz representations up to a maximal spin $J_{\max}$ is $$ N_{\text{Lorentz}}(J_{\max})=\sum_{m=0}^{J_{\max}}\sum_{n=0}^{J_{\max}} \bigl[1+\delta_{m\neq n}\bigr] = (J_{\max}+1)^{2} + \bigl[(J_{\max}+1)^{2}-(J_{\max}+1)\bigr] =2(J_{\max}+1)^{2}-(J_{\max}+1). $$ Even for $J_{\max}=10$ this yields $N_{\text{Lorentz}}(10)=2\cdot11^{2}-11=242-11=231$, already far exceeding the finite bound $|G|=24$ of the $SL(2,3)$ model.

   from 2. Infinite Lorentz towers versus finite‑group sectors

6. Applying the result to the concrete candidate $G=SL(2,3)$ yields a hard bound of five distinct charge sectors, far below the infinite spectrum required by the Standard Model’s continuous gauge group.

   from 9. Conclusion

How it is done

1. We combine (i) the classification of finite‑group 2‑gauge theories [2], (ii) the excess‑zero property of finite Coxeter groups [1], (iii) the density‑of‑states formula for finite‑group symmetry [3], and (iv) the factorisation of gauge symmetry on intervals [4] to prove a no‑go theorem.

   from Abstract

2. Our contributions are: Theorem 1 (Section 4) – A rigorous proof that any lattice TQFT whose underlying braided fusion category has a finite simple‑object set can only realise a finite gauge symmetry in the continuum limit.

   from 1. Introduction

3. Our contribution is to synthesize these observations into a rigorous no‑go theorem.

   from 2. Background and Related Work

4. Our analysis proceeds in three stages.

   from 3. Methods

What would falsify it

1. The main claim would be falsified if a concrete lattice model built from a finite‑object braided fusion category were shown to possess an infinite tower of distinct bulk excitations in the continuum limit, or if such a model reproduced the full SM anomaly cancellation pattern with only finitely many charge sectors.

   from 7.2. Potential falsification

2. An explicit counterexample would require either (i) a new categorical construction that yields infinitely many simple objects from a finite underlying group, or (ii) a demonstration that the renormalisation‑group flow generates emergent continuous symmetry without adding new simple objects.

   from 7.2. Potential falsification

3. The observable is the number of distinct topological charge sectors.

   from (c) Minimal next experiment or prototype

Where the claim stops

1. Assumption Sensitivity A1 – $|SL(2,3)|=24$ and five conjugacy classes (standard finite‑group fact).

   from 7.1. Limitations

2. A3 – Continuum limit does not introduce new superselection sectors beyond those of the lattice model.

   from 7.1. Limitations

3. If a novel RG fixed point generated emergent continuous symmetry, the proof would need extension; however, no such mechanism is known for finite‑group TQFTs.

   from 7.1. Limitations

Where to read

1. Abstract202 words · 1 min
2. 1. Introduction330 words · 1 min
3. 2. Background and Related Work497 words · 2 min
4. 3. Methods194 words · 1 min
5. 4. Analysis646 words · 3 min
6. 5. Results167 words · 1 min
7. Appendix C. Extended Derivations and Worked Examples1,083 words · 5 min
8. 7. Discussion264 words · 1 min
9. 7. Practical Implications and Engineering Pathways318 words · 1 min
10. 9. Conclusion122 words · 1 min
11. Provenance and AI Involvement225 words · 1 min