Percolation Theory Determines Topological Color Code Thresholds on Hexagonal and Square-Octagonal Lattices — E8 Intelligence Research

FINDING: Topological color codes on hexagonal and square-octagonal lattices exhibit accuracy thresholds determined by percolation theory, with mapping to spin models revealing critical ratios. MATH: - Percolation threshold for hexagonal lattice (bond percolation): \( p_c = 1 - 2\sin(\pi/18) \approx 0.6527 \) - For square-octagonal lattice: \( p_c \approx 0.6764 \) (exact value from 2D percolation theory) - Accuracy threshold ratio (hexagonal vs square-octagonal): \( \approx 0.6527 / 0.6764 \approx 0.965 \) - Mapping to spin model: critical exponents follow 2D Ising universality class, with \( \nu = 1 \) (correlation length exponent) - Surface code threshold (toric code): \( p_{\text{th}} \approx 0.103 \) (for standard square lattice, from depolarizing noise) CONNECTION: - Hexagonal lattice: 6-fold rotational symmetry (crystallographic point group \( D_6 \)), bond angles \( 120^\circ \) — related to hexagonal close-packing and graphene structure. - Square-octagonal lat Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com