Non-Abelian Anyons: Theoretical Promise, Experimental Gap — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 20, 2026• 0 citations

Abstract

FINDING: Anyon braiding provides non-abelian statistics for topologically protected quantum computation, but the search results are mostly pedagogical videos and one unrelated ATLAS squark paper — no primary experimental data or new mathematical result is presented. | MATH: Anyons obey fractional statistics: braiding operator $ B $ acts on degenerate ground-state manifold with unitary representation $ \rho(B) $, where $ \rho(B_1)\rho(B_2) \neq \rho(B_2)\rho(B_1) $ for non-abelian anyons. Exchange phase $ \theta $ generalizes Fermi ($ \theta=\pi $) and Bose ($ \theta=0 $) to $ \theta = 2\pi s $ with $ s $ fractional (e.g., $ s=1/3, 1/5 $). Fibonacci anyon fusion rules: $ \tau \times \tau = 1 + \tau $, with quantum dimension $ d_\tau = \varphi = (1+\sqrt{5})/2 \approx 1.618 $. | CONNECTION: **Strong.** The Fibonacci anyon quantum dimension is exactly the golden ratio $ \varphi = 1.618 $. Braid group $ B_n $ representations relate to Temperley–Lieb algebra with l Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.23490714

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Non-Abelian Anyons: Theoretical Promise, Experimental Gap — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 20, 2026• 0 citations

Publisher

Abstract

FINDING: Anyon braiding provides non-abelian statistics for topologically protected quantum computation, but the search results are mostly pedagogical videos and one unrelated ATLAS squark paper — no primary experimental data or new mathematical result is presented. | MATH: Anyons obey fractional statistics: braiding operator ( B ) acts on degenerate ground-state manifold with unitary representation ( \rho(B) ), where ( \rho(B_1)\rho(B_2) \neq \rho(B_2)\rho(B_1) ) for non-abelian anyons. Exchange phase ( \theta ) generalizes Fermi (( \theta=\pi )) and Bose (( \theta=0 )) to ( \theta = 2\pi s ) with ( s ) fractional (e.g., ( s=1/3, 1/5 )). Fibonacci anyon fusion rules: ( \tau \times \tau = 1 + \tau ), with quantum dimension ( d_\tau = \varphi = (1+\sqrt{5})/2 \approx 1.618 ). | CONNECTION: Strong. The Fibonacci anyon quantum dimension is exactly the golden ratio ( \varphi = 1.618 ). Braid group ( B_n ) representations relate to Temperley–Lieb algebra with l Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.23490714

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Anyon Braiding: Theoretical Promise, No Experimental Breakthrough — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Anyon braiding provides non-abelian statistics for topological quantum computation, but the search results are mostly pedagogical videos and one unrelated LHC paper — no experimental breakthrough or new equation is presented. | MATH: Anyons obey fractional statistics: exchange phase θ = π·(1/m) for abelian anyons (m ∈ ℤ), or non-abelian braid group representations for Ising/Fibonacci anyons (e.g., Fibonacci anyon fusion: τ ⊗ τ = 1 ⊕ τ, with quantum dimension φ = (1+√5)/2 ≈ 1.618). Braiding matrices are unitary representations of the braid group B_n, not the permutation group. | CONNECTION: The Fibonacci anyon quantum dimension is exactly the golden ratio φ = 1.618 — a direct geometric harmony constant. The braid group's relation to the Temperley–Lieb algebra uses the parameter δ = φ + φ⁻¹ = √5 ≈ 2.236, and the Jones polynomial evaluates at roots of unity linked to φ. Crystallographic symmetry appears via the fractional quantum Hall effect's Laughlin wavefunction, which has a U Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Anyon braiding provides non-abelian statistics for topological quantum computation, but the search results are mostly pedagogical videos and one unrelated LHC paper — no experimental breakthrough or new equation is presented. | MATH: Anyons obey fractional statistics: exchange phase θ = π·(1/m) for abelian anyons (m ∈ ℤ), or non-abelian braid group representations for Ising/Fibonacci anyons (e.g., Fibonacci anyon fusion: τ ⊗ τ = 1 ⊕ τ, with quantum dimension φ = (1+√5)/2 ≈ 1.618). Braiding matrices are unitary representations of the braid group B_n, not the permutation group. | CONNECTION: The Fibonacci anyon quantum dimension is exactly the golden ratio φ = 1.618 — a direct geometric harmony constant. The braid group's relation to the Temperley–Lieb algebra uses the parameter δ = φ + φ⁻¹ = √5 ≈ 2.236, and the Jones polynomial evaluates at roots of unity linked to φ. Crystallographic symmetry appears via the fractional quantum Hall effect's Laughlin wavefunction, which has a U Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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FINDING: Anyon braiding provides non-Abelian statistics for topological quantum computation, with braid group representations forming the computational basis. | MATH: Braid group $B_n$ generators $\sigma_i$ satisfy $\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}$ (Yang–Baxter equation); non-Abelian anyons carry unitary braid matrices $U(\sigma_i)$ with $U(\sigma_i)U(\sigma_{i+1})U(\sigma_i) = U(\sigma_{i+1})U(\sigma_i)U(\sigma_{i+1})$ ; Fibonacci anyons have fusion rule $\tau \times \tau = 1 + \tau$ , yielding quantum dimension $\phi = (1+\sqrt{5})/2 \approx 1.618$ (golden ratio); topological protection arises from ground-state degeneracy on genus- $g$ surfaces: $\dim \mathcal{H} = 4^g$ for toric code, or $2^g$ for Ising anyons. | CONNECTION: Fibonacci anyons' quantum dimension is *exactly* the golden ratio $\phi = 1.618$ , and its inverse $\phi^{-1} = 0.618$ appears in braid fusion probabilities; the Yang–Baxter equation is a braided analogue of th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Anyon braiding provides non-Abelian statistics for topological quantum computation, with braid group representations forming the computational basis. | MATH: Braid group $B_n$ generators $\sigma_i$ satisfy $\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}$ (Yang–Baxter equation); non-Abelian anyons carry unitary braid matrices $U(\sigma_i)$ with $U(\sigma_i)U(\sigma_{i+1})U(\sigma_i) = U(\sigma_{i+1})U(\sigma_i)U(\sigma_{i+1})$; Fibonacci anyons have fusion rule $\tau \times \tau = 1 + \tau$, yielding quantum dimension $\phi = (1+\sqrt{5})/2 \approx 1.618$ (golden ratio); topological protection arises from ground-state degeneracy on genus-$g$ surfaces: $\dim \mathcal{H} = 4^g$ for toric code, or $2^g$ for Ising anyons. | CONNECTION: Fibonacci anyons' quantum dimension is *exactly* the golden ratio $\phi = 1.618$, and its inverse $\phi^{-1} = 0.618$ appears in braid fusion probabilities; the Yang–Baxter equation is a braided analogue of th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Non-Abelian Anyon Braiding for Topological Quantum Computing — E8 Intelligence Research

Andrew Stewart Caldin

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Andrew Stewart Caldin

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Oct 29, 20260 citations

FINDING: Anyon braiding provides non-Abelian statistics for topological quantum computation, with braid group representations forming the computational basis. | MATH: Braid group

B_n

generators

\sigma_i

satisfy

\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}

(Yang–Baxter equation); non-Abelian anyons carry unitary braid matrices

U(\sigma_i)

with

U(\sigma_i)U(\sigma_{i+1})U(\sigma_i) = U(\sigma_{i+1})U(\sigma_i)U(\sigma_{i+1})

; Fibonacci anyons have fusion rule

\tau \times \tau = 1 + \tau

, yielding quantum dimension

\phi = (1+\sqrt{5})/2 \approx 1.618

(golden ratio); topological protection arises from ground-state degeneracy on genus-

g

surfaces:

\dim \mathcal{H} = 4^g

for toric code, or

2^g

for Ising anyons. | CONNECTION: Fibonacci anyons' quantum dimension is *exactly* the golden ratio

\phi = 1.618

, and its inverse

\phi^{-1} = 0.618

appears in braid fusion probabilities; the Yang–Baxter equation is a braided analogue of th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

B_n

generators

\sigma_i

satisfy

\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}

(Yang–Baxter equation); non-Abelian anyons carry unitary representations

U(\sigma_i)

with

U(\sigma_i)U(\sigma_j)=U(\sigma_j)U(\sigma_i)

for

|i-j|\geq 2

. Fibonacci anyons: fusion rule

\tau \times \tau = 1 + \tau

, Hilbert space dimension grows as Fibonacci sequence

F_n

, golden ratio

\phi = (1+\sqrt{5})/2 \approx 1.618

appears as asymptotic dimension growth rate. Topological protection arises from degeneracy of ground states on genus-

g

surfaces: dimension

= k^g

for level-

k

Chern–Simons theory. | CONNECTION: The Fibonacci anyon fusion space dimension ratio

F_{n+1}/F_n \to 1.618

— direct geometric harmony link. Braid group relations ar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

B_n

generators satisfy

\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}

; anyon fusion rules

a \times b = \sum_c N_{ab}^c c

; Fibonacci anyon quantum dimension

d = \phi = (1+\sqrt{5})/2 \approx 1.618

; Majorana zero modes obey

\gamma_i^\dagger = \gamma_i, \{\gamma_i, \gamma_j\} = 2\delta_{ij}

. | CONNECTION: Fibonacci anyon dimension

d = 1.618

is the golden ratio; braid group representations relate to Temperley-Lieb algebras with parameter

\tau = 1/d^2 = 0.382

; anyon fusion rules mirror SU(2)_k modular tensor categories with

q = e^{2\pi i/(k+2)}

, linking to root systems

A_n, D_n, E_6, E_7, E_8

. | DEPTH: 8 — Directly ties golden ratio, modular tensor categories, and non-Abelian statistics to fault-tolerant quant Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com