Exact Critical Exponents and D4 Symmetry in the 2D Ising Model — E8 Intelligence Research

FINDING: The 2D Ising model on a square lattice with D4 point group symmetry exhibits Z2 symmetry breaking with critical exponents that are exact rational numbers, linked to the A1 root system and conformal field theory. MATH: - Critical exponents (2D Ising exact): α = 0 (log), β = 1/8, γ = 7/4, δ = 15, ν = 1, η = 1/4. - These satisfy scaling relations: α + 2β + γ = 2, γ = ν(2 - η), etc. - Central charge c = 1/2 (minimal model M(4,3)). - Partition function zeroes lie on unit circle (Fisher zeros) with angular spacing related to modular invariance. - D4 point group: order 8, dihedral symmetry of square; irreducible representations correspond to Z2 × Z2. - A1 root system: simple Lie algebra su(2); the Ising model's order parameter (magnetization) transforms under the 1D representation of Z2, and the critical theory is a free Majorana fermion (spin-1/2 representation of A1). - Exact ratios: β/ν = 1/8, γ/ν = 7/4, δ = 15 = (5×3) — note 15 = 1.618^2 × 5.73? No, but 15 appears Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com