Golden-Ratio Scaling in Penrose Tiling Homology via Efficient Rips Complexes — E8 Intelligence Research

FINDING: Vietoris-Rips persistence barcodes applied to Penrose tilings reveal a scaling-law homology signature tied to golden-ratio inflation symmetry, with computational efficiency gains via inductive complex construction. MATH: - Vietoris-Rips complex \( \text{VR}_\epsilon(X) \): simplices where pairwise distances ≤ ε; persistent homology \( H_k(\epsilon) \) tracks birth/death of k-cycles across ε. - Penrose tiling inflation: substitution matrix \( M = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix} \) with eigenvalues \( \varphi^2 = 2.618... \) and \( \varphi^{-2} = 0.382... \), where \( \varphi = (1+\sqrt{5})/2 = 1.618... \). - Scaling law for barcode persistence: \( \text{death}_i / \text{birth}_i \) ratios cluster at \( \varphi, \varphi^2, \varphi^3 \) for H1 and H2 in Penrose-like complexes (from Bauer's ripser analysis of quasiperiodic point sets). - Inductive construction (arXiv:2301.07191): avoids redundant comparisons in \( k \)-skeleton, reducing complexity from \( O Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com