A Taxonomy of Exchange Statistics from Configuration-Space Topology: Non-Abelian Flux Attachment and a Pre-Registered Thermal Hall Projection

In the paper's words

The statistics of identical particles is fixed by the fundamental group of their configuration space: in d ≥ 3 spatial dimensions this group is the symmetric group S_N, permitting only the bosonic (+1) and fermionic (-1) exchange phases, while in $d = 2$ it is the braid group B_N, whose one-dimensional representations give anyons and whose higher-dimensional representations give non-Abelian anyons. What is missing is a systematic classification of the exchange behaviors obtained by attaching fluxes drawn from a possibly non-Abelian group G to particle worldlines, organized so that each class carries a falsifiable observable. This paper develops such a taxonomy as a four-axis classification: (A) the exchange group G_N = π₁(C_N(M)); (B) the dimension r of the unitary representation ρ: G_N → U(r); (C) whether ρ factors through a finite quotient; and (D) the chiral central charge c_- of the associated topological order.

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

Numbers it reports

1. We derive the abelianization $B_N^{\mathrm{ab}} \cong \mathbb{Z}$, compute the thermal Hall conductance quantum $\pi^2 k_B^2/(3h) = 9.464311516 \times 10^{-13}\ \mathrm{W\,K^{-2}}$ from the exact 2019-SI values of $k_B$ and $h$, and project $\kappa_{xy}/T = 1.419646727 \times 10^{-12}\ \mathrm{W\,K^{-2}}$ for a candidate non-Abelian class with $c_- = 3/2$, i.e. $\kappa_{xy} = 1.419646727 \times 10^{-14}\ \mathrm{W\,K^{-1}}$ at $T = 10\ \mathrm{mK}$ — a factor-of-3 discriminator against the Ising-type $c_- = 1/2$ alternative.

   from Abstract

2. The contributions of this paper are: A four-axis taxonomy $(G_N, r, \mathrm{factor}, c_-)$ of exchange statistics under homomorphism-defined flux attachment, with the Abelian/non-Abelian and finite-factor/genuinely-braided distinctions made explicit (Section 3.3).

   from 1. Introduction

3. Numerical family predictions $\kappa_{xy}/T$ for $c_- \in \{1/2, 1, 3/2, 5/2\}$ and the factor-of-3 discriminator between the $c_- = 3/2$ and $c_- = 1/2$ classes at $T = 10\ \mathrm{mK}$ (Section 4.4).

   from 1. Introduction

4. Square $k_B$: $$k_B^2 = (1.380649)^2 \times 10^{-46} = 1.906191661 \times 10^{-46}\ \mathrm{J^2\,K^{-2}}.$$ Explicitly, $1.380649 \times 1.380649$: $1.380649 \times 1 = 1.380649$; $1.380649 \times 0.3 = 0.4141947$; $1.380649 \times 0.08 = 0.11045192$; $1.380649 \times 0.0006 = 0.0008283894$; $1.380649 \times 0.00004 = 0.0000552260$; $1.380649 \times 0.000009 = 0.0000124258$; summing the partial products for the second factor digit by digit gives $1.906191661$ (to nine decimal places).

   from 4.3 Axis (D): the thermal Hall conductance quantum

5. The ratio of the $c_- = 3/2$ slope to the $c_- = 1/2$ slope is $1.5/0.5 = 3$: a factor-of-3 separation between the candidate non-Abelian class and the Ising-type alternative, far larger than plausible systematic errors in a quantized measurement, so a single slope measurement with modest precision discriminates the two classes.

   from 4.4 Candidate chiral central charges and numerical signatures

6. The enlargement factor is $m/2$ (e.g., $1.5$ for $m = 3$, $5$ for $m = 10$), quantifying the genuine two-dimensional richness of the taxonomy.

   from 4.5 Distinguishing power against the boson/fermion shadow

How it is done

1. We derive the abelianization $B_N^{\mathrm{ab}} \cong \mathbb{Z}$, compute the thermal Hall conductance quantum $\pi^2 k_B^2/(3h) = 9.464311516 \times 10^{-13}\ \mathrm{W\,K^{-2}}$ from the exact 2019-SI values of $k_B$ and $h$, and project $\kappa_{xy}/T = 1.419646727 \times 10^{-12}\ \mathrm{W\,K^{-2}}$ for a candidate non-Abelian class with $c_- = 3/2$, i.e. $\kappa_{xy} = 1.419646727 \times 10^{-14}\ \mathrm{W\,K^{-1}}$ at $T = 10\ \mathrm{mK}$ — a factor-of-3 discriminator against the Ising-type $c_- = 1/2$ alternative.

   from Abstract

2. This dimension-dependent dichotomy is a theorem about the topology of the space in which particles move, as the configuration-space-topology (CST) framework states [10]; the same framework acknowledges that it rests on scaffolds — point-like particles, a fixed classical background manifold, a deleted hard-core diagonal, an externally imposed dimension — that it cannot itself derive [10].

   from 1. Introduction

3. This work determines the structure of the $n$-dimensional Hawaiian group of the $m$-dimensional Hawaiian earring space for all $1 \leq m \leq n$, and studies Hawaiian groups under product spaces, weak join spaces, cone spaces, covering spaces, and locally trivial bundles [3]; its Theorem 3.1 gives, for a first countable, locally strongly contractible pointed space, an isomorphism of the infinite-dimensional Hawaiian group onto a weak product of ordinary homotopy groups [3].

   from 2. Background and Related Work

4. Each family in the taxonomy predicts a value of $c_-$ via its flux-fusion structure; we compute these in Section 4.

   from 3.4 Thermal Hall conductance as the observable

What would falsify it

1. What is missing is a systematic classification of the exchange behaviors obtained by attaching fluxes drawn from a possibly non-Abelian group $G$ to particle worldlines, organized so that each class carries a falsifiable observable.

   from Abstract

2. Assumptions, failure modes, and falsification criteria are stated explicitly, and the result gives low-temperature thermal-transport experiments a concrete quantized target against which candidate topological orders can be confirmed or excluded.

   from Abstract

3. A measurement campaign that cannot name its target slope cannot distinguish confirmation of a candidate order from refutation of it; the cost is misassigned topological orders and, downstream, blocked progress toward topological quantum computation, where non-Abelian anyons are the resource.

   from 1. Introduction

Where the claim stops

1. Thermodynamic or continuum limits of $N$ require coarse-structure machinery of the kind developed for minimal topological groups in Ref. [7]; we do not claim the discrete taxonomy survives that limit without modification.

   from 6. Discussion

2. Axes (A)–(C) do not determine axis (D): the same braid representation can be realized by modular tensor categories with different $c_-$.

   from 6. Discussion

3. Any claim that braid-group data alone fixes the thermal Hall response is incorrect, and this paper explicitly does not make it.

   from 6. Discussion

Where to read

1. Abstract243 words · 1 min
2. 1. Introduction633 words · 3 min
3. 2. Background and Related Work1,368 words · 6 min
4. 3. Methods499 words · 2 min
5. 4. Analysis834 words · 4 min
6. 5. Results476 words · 2 min
7. 6. Discussion644 words · 3 min
8. 7. Practical Implications and Engineering Pathways473 words · 2 min
9. 8. Conclusion280 words · 1 min
10. Provenance and AI Involvement225 words · 1 min