D₃ as Minimal Non-Abelian Group for Quantum Ternary Choice on Tianyan-287 — E8 Intelligence Research

Abstract

FINDING: Dihedral group D₃ (symmetry group of equilateral triangle) is the minimal non-abelian group, directly relevant to modeling non-commutative ternary choice operators in quantum cognition; the Tianyan-287 quantum processor (105 qubits) provides a physical platform for testing such non-abelian group actions in superposition. | MATH: D₃ has order 6, presentation ⟨r, s | r³ = s² = e, srs = r⁻¹⟩. Non-commutativity: sr ≠ rs. Irreducible representations: two 1D (trivial, sign) and one 2D (faithful, with rotation matrix R(2π/3) = [[-1/2, -√3/2], [√3/2, -1/2]] and reflection S = [[1, 0], [0, -1]]). The 2D irrep's character values: χ(e)=2, χ(r)=−1, χ(s)=0. Ternary choice operators in quantum cognition can be represented as elements of D₃ acting on a 3-dimensional Hilbert space basis {|A⟩, |B⟩, |C⟩}, where r cyclically permutes (A→B→C→A) and s swaps two options while fixing the third. | CONNECTION: The 2D irrep of D₃ contains the golden ratio implicitly: eigenvalues of R(2π/3) are e^{±2πi/ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com