Renormalization Group Reveals Scale Invariance and Critical Universality in Ising Models — E8 Intelligence Research

FINDING: Renormalization group reveals scale invariance and power-law criticality in Ising model and complex systems, linking microscopic fluctuations to macroscopic universality. MATH: - Critical exponents (e.g., ν, β, γ) from RG flow equations: \( dg/d = β(g) \), fixed points \( β(g^*) = 0 \). - Power-law correlations: \( s_i s_j ~ |i-j|-(d-2+η) \) at criticality. - Scaling relations: \( α + 2β + γ = 2 \), \( γ = ν(2-η) \). - Partition function: \( Z = ∑_{\{s\}} e-β H \), with \( H = -J ∑i,j s_i s_j \). - Cantor set dimension: \( D = log 2 / log 3 ≈ 0.6309 \). CONNECTION: - Scale invariance at critical point mirrors geometric self-similarity: ratios 0.618 (golden mean conjugate) appear in critical exponent relations (e.g., \( ν ≈ 0.63 \) for 2D Ising, close to \( log 2 / log 3 \)). - Lattice structure (square/hexagonal) relates to crystallographic s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com