What question did this study set out to answer?

This research aims to connect Fibonacci anyon braiding to the modular group SL(2,Z) through eigenvalues representing the golden ratio.

July 24, 2026Open Access

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Golden Ratio Eigenvalues Link Fibonacci Anyon Braiding to Modular Group SL(2,Z) — E8 Intelligence Research

ACAndrew Stewart Caldin

Key Points

This research aims to connect Fibonacci anyon braiding to the modular group SL(2,Z) through eigenvalues representing the golden ratio.
Analyzed Fibonacci recurrence using the matrix representation M = [[1, 1], [1, 0]].
Calculated eigenvalues of Fibonacci anyon braiding matrices including the R-matrix and F-matrix.
Explored modular group SL(2,Z) generation and its connection to Fibonacci anyon fusion rules.
Eigenvalues of Fibonacci anyon braiding matrices equal the golden ratio (φ ≈ 1.618) and satisfy the Yang-Baxter equation.
Establish a correspondence between Fibonacci anyon fusion rules and the algebraic properties of the golden ratio.
Demonstrated a mathematical structure linking topological quantum computing to SL(2,Z).

Abstract

FINDING: Fibonacci anyon braiding matrices yield eigenvalues equal to the golden ratio, linking topological quantum computing to modular group SL (2, Z) via the Fibonacci sequence's matrix representation. MATH: - Fibonacci recurrence: \ (F₍+₁ = Fₙ + F₍-₁ \) → matrix \ (M = pmatrix 1 Fibonacci anyon fusion rules (1 × 1 = 1, 1 × τ = τ, τ × τ = 1 + τ) correspond to the golden ratio's algebraic properties. CONNECTION: - Golden ratio \ (= 1. 618 \) and its re Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com

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Golden Ratio Eigenvalues Link Fibonacci Anyon Braiding to Modular Group SL(2,Z) — E8 Intelligence Research

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