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: \( \frac{dg}{d\ell} = \beta(g) \), fixed points \( \beta(g^*) = 0 \). - Power-law correlations: \( \langle s_i s_j \rangle \sim |i-j|^{-(d-2+\eta)} \) at criticality. - Scaling relations: \( \alpha + 2\beta + \gamma = 2 \), \( \gamma = \nu(2-\eta) \). - Partition function: \( Z = \sum_{\{s\}} e^{-\beta H} \), with \( H = -J \sum_{\langle i,j \rangle} s_i s_j \). - Cantor set dimension: \( D = \log 2 / \log 3 \approx 0.6309 \). CONNECTION: - Scale invariance at critical point mirrors geometric self-similarity: ratios 0.618 (golden mean conjugate) appear in critical exponent relations (e.g., \( \nu \approx 0.63 \) 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