E8 in Condensed Matter: From Ising Chains to Quantum Hall States and Fibonacci Anyons — Three Experimental Windows on the E8 Root Lattice

Survey and synthesis of the three known appearances of the exceptional Lie group E8 in condensed-matter physics, with every claim classified by epistemic status (derived / conditional / open). (1) Ising chains: the E8 Toda field theory of the critical Ising chain in a longitudinal field (Zamolodchikov 1989). The 8 mass ratios follow from the Perron–Frobenius eigenvector of the E8 Cartan matrix; they are re-verified here to 9 decimal places. Experimental confirmations in CoNb2O6, BaCo2V2O8, Cu2(OH)3Br and a Rydberg-atom quantum simulator are tabulated, with percent-level agreement in the best-measured ratios. (2) Quantum Hall: the bosonic E8 state whose Chern–Simons K-matrix is the E8 Cartan matrix — the unique 8×8 symmetric, unimodular, positive-definite, even integer matrix (Kitaev; Lu–Vishwanath). Falsifiable prediction: thermal Hall conductance κ_xy = 8 κ0 (PRED-063). (3) Fibonacci anyons: the conformal embedding (E8)_1 = (G2)_1 × (F4)_1 (central charges 14/5 + 26/5 = 8) yields the Fibonacci anyon, universal for topological quantum computation; edge-tunneling prediction P7 = h_G2/h_F4 = 2/3 vs. null hypothesis 7/13 (PRED-075). Structural bridge: the centralizer of the Z_3B conjugacy class of E8 is E6 × SU(3) — simultaneously the particle-physics branching used in the TTOE framework and the control element for Fibonacci/Haagerup symmetry in (E8)_1. The paper states explicitly what is NOT claimed: E8 in condensed matter is emergent, and its appearance is explained by standard condensed-matter theory without any fundamental-physics framework. Version 02 (24 August 2026): bibliography fully verified and cited from the text; corrected the scaling-exponent identity to 8/15 = 2·rk(E8)/h∨; experimental-precision claims aligned with the sources.