Fibonacci Anyon Braid Groups Classified by Modular Tensor Categories — E8 Intelligence Research

Andrew Stewart Caldin

Published Jul 23, 2026• 0 citations

Abstract

FINDING: Fibonacci anyon braid group representations are classified by modular tensor categories (MTCs) with fusion rules derived from the Fibonacci (golden ratio) quantum group $ \text{SU}(2)_3 $, yielding non-Abelian statistics and universal topological quantum computation. MATH: - Fibonacci anyon fusion rule: $ \tau \times \tau = 1 + \tau $ (where $ \tau $ is the non-Abelian anyon, 1 is the vacuum). - Quantum dimension of $ \tau $: $ d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618 $. - Braid matrix eigenvalues: $ e^{\pm i\pi/5} $ and $ e^{\pm i3\pi/5} $, derived from the $ R $-matrix of the quantum group $ U_q(\mathfrak{sl}_2) $ at $ q = e^{i\pi/5} $. - Modular $ S $-matrix for Fibonacci MTC: \[ S = \frac{1}{\sqrt{\phi+2}} \begin{pmatrix} 1 & \phi \\ \phi & -1 \end{pmatrix} \] with $ \phi = 1.618 $. - Central charge $ c = 4/5 $, related to the minimal model $ M(5,4) $ of conformal field theory. CONNECTION: - The golden ratio \( \phi = Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.21503632

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Fibonacci Anyon Braid Groups Classified by Modular Tensor Categories — E8 Intelligence Research

Andrew Stewart Caldin

Published Jul 23, 2026• 0 citations

Publisher

Abstract

FINDING: Fibonacci anyon braid group representations are classified by modular tensor categories (MTCs) with fusion rules derived from the Fibonacci (golden ratio) quantum group ( \text{SU}(2)3 ), yielding non-Abelian statistics and universal topological quantum computation. MATH: - Fibonacci anyon fusion rule: ( \tau \times \tau = 1 + \tau ) (where ( \tau ) is the non-Abelian anyon, 1 is the vacuum). - Quantum dimension of ( \tau ): ( d\tau = \phi = (1+\sqrt{5})/2 \approx 1.618 ). - Braid matrix eigenvalues: ( e^{\pm i\pi/5} ) and ( e^{\pm i3\pi/5} ), derived from the ( R )-matrix of the quantum group ( U_q(\mathfrak{sl}_2) ) at ( q = e^{i\pi/5} ). - Modular ( S )-matrix for Fibonacci MTC: [ S = \frac{1}{\sqrt{\phi+2}} \begin{pmatrix} 1 & \phi \ \phi & -1 \end{pmatrix} ] with ( \phi = 1.618 ). - Central charge ( c = 4/5 ), related to the minimal model ( M(5,4) ) of conformal field theory. CONNECTION: - The golden ratio ( \phi = Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.21503632

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Fibonacci Anyons: Golden Ratio Fusion Rules Link Quantum Computing to Ancient Harmony — E8 Intelligence Research

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FINDING: Fibonacci anyon fusion rules are governed by the golden ratio φ, with quantum dimension d = φ = (1+√5)/2, and the modular tensor category (MTC) classification reveals that the Fibonacci category is the simplest non-abelian anyon model, directly linking topological quantum computation to ancient geometric harmony. MATH: - Quantum dimension: $d = \phi = \frac{1+\sqrt{5}}{2} \approx 1.618$ - Fusion rule: $\tau \times \tau = 1 + \tau$ (single non-trivial anyon type τ) - Modular S-matrix element: $S_{\tau\tau} = \frac{1}{\sqrt{1+\phi^2}} = \frac{1}{\sqrt{2+\phi}}$ - Central charge: $c = 0 \mod 8$ (for Fibonacci MTC) - Key constants: φ, 1/φ ≈ 0.618, φ² ≈ 2.618, φ⁻² ≈ 0.382 CONNECTION: - Golden ratio φ = 1.618 appears directly as quantum dimension; its reciprocal 0.618 and square 2.618 are inherent in fusion probabilities and braiding phases. - The ratio 0.382 = φ⁻² emerges in the probability amplitude for fusion outcomes. - The Fibonacci MTC is re Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Fibonacci anyon fusion rules are governed by the golden ratio φ, with quantum dimension d = φ = (1+√5)/2, and the modular tensor category (MTC) classification reveals that the Fibonacci category is the simplest non-abelian anyon model, directly linking topological quantum computation to ancient geometric harmony. MATH: - Quantum dimension: $ d = \phi = \frac{1+\sqrt{5}}{2} \approx 1.618 $ - Fusion rule: $ \tau \times \tau = 1 + \tau $ (single non-trivial anyon type τ) - Modular S-matrix element: $ S_{\tau\tau} = \frac{1}{\sqrt{1+\phi^2}} = \frac{1}{\sqrt{2+\phi}} $ - Central charge: $ c = 0 \mod 8 $ (for Fibonacci MTC) - Key constants: φ, 1/φ ≈ 0.618, φ² ≈ 2.618, φ⁻² ≈ 0.382 CONNECTION: - Golden ratio φ = 1.618 appears directly as quantum dimension; its reciprocal 0.618 and square 2.618 are inherent in fusion probabilities and braiding phases. - The ratio 0.382 = φ⁻² emerges in the probability amplitude for fusion outcomes. - The Fibonacci MTC is re Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Andrew Stewart Caldin

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Oct 18, 20260 citations

d = \phi = \frac{1+\sqrt{5}}{2} \approx 1.618

- Fusion rule:

\tau \times \tau = 1 + \tau

(single non-trivial anyon type τ) - Modular S-matrix element:

S_{\tau\tau} = \frac{1}{\sqrt{1+\phi^2}} = \frac{1}{\sqrt{2+\phi}}

- Central charge:

c = 0 \mod 8

(for Fibonacci MTC) - Key constants: φ, 1/φ ≈ 0.618, φ² ≈ 2.618, φ⁻² ≈ 0.382 CONNECTION: - Golden ratio φ = 1.618 appears directly as quantum dimension; its reciprocal 0.618 and square 2.618 are inherent in fusion probabilities and braiding phases. - The ratio 0.382 = φ⁻² emerges in the probability amplitude for fusion outcomes. - The Fibonacci MTC is re Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

d_\tau = \phi = \frac{1+\sqrt{5}}{2} \approx 1.618

. - Braiding matrix (R-matrix) for two τ anyons: eigenvalues are

e^{\pm i 4\pi/5}

, related to

\phi

via

2\cos(2\pi/5) = \phi^{-1} = 0.618

. - Unitary braid group representations generated by matrices satisfying

B^2 = \phi^{-1} B + \phi^{-2} I

. CONNECTION: - Golden ratio φ appears as quantum dimension and in braid eigenvalues; 0.618 = φ⁻¹ emerges from 5-fold rotational symmetry (pentagon, icosahedron). - Base-60 not directly present, but 5-fold symmetry links to crystallographic restrictions (Penrose tilings, quasicrystals). - Braiding statistics correspond to the Fibonacci category, a modular tenso Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

N^k_{ij}

S

-matrix (modular transformation

\tau \to -1/\tau

T

-matrix (diagonal,

\theta_a = e^{2\pi i h_a}

topological spins), satisfying

S^2 = C

(charge conjugation),

(ST)^3 = e^{2\pi i c/8} S^2

(where

c

is the chiral central charge). The entries of

S

and

T

lie in a cyclotomic field

\mathbb{Q}(\zeta_N)

\zeta_N = e^{2\pi i/N}

. For the Fibonacci anyon MTC:

S = \frac{1}{\sqrt{\phi+2}}\begin{pmatrix} 1 & \phi \\ \phi & -1 \end{pmatrix}

T = \text{diag}(1, e^{4\pi i/5})

, where

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

. | CONNECTION: **Direct hit**: The Fibonacci MTC's

S

-matrix contains

\phi = 1.618

and

\phi+2 = \phi^2+1 = 3.618

, whose inverse square root yields

1/\sqrt{3.618} \approx 0.5257

. The golden ratio appears intrinsically. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

FINDING: Non-Abelian anyon fusion algebra exhibits self-similarity under braid group representations, enabling topological quantum computation via braiding operations. MATH: - Braid group

B_n

generators

\sigma_i

satisfy

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

and

\sigma_i \sigma_j = \sigma_j \sigma_i

for

|i-j|>1

. - Hecke algebra relations:

(\sigma_i - q^{1/2})(\sigma_i + q^{-1/2}) = 0

, with

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

for SU(2)_k anyons. - Fusion rules:

\phi_a \times \phi_b = \sum_c N_{ab}^c \phi_c

, where

N_{ab}^c

are non-negative integers (fusion coefficients). - Fibonacci anyon fusion:

\tau \times \tau = 1 + \tau

, with quantum dimension

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

. CONNECTION: - Fibonacci anyon quantum dimension

d_\tau = \phi = 1.618

(golden ratio). - Braid matrix eigenvalues for Fibonacci anyons:

e^{\pm 3\pi i/5}

, related to golden ratio via \( 2\cos(3\pi/5) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com