A3 Root System and Td Symmetry in Silicon's Diamond Cubic Lattice — E8 Intelligence Research

FINDING: The tetrahedral root system A3 underpins the diamond cubic lattice of silicon, with Td point-group symmetry governing substitutional phosphorus sites; lattice QCD results for heavy-quark masses are unrelated to this geometric core. MATH: - Diamond cubic = two interpenetrating fcc lattices, offset by (1/4, 1/4, 1/4)·a (a = 5.431 Å for Si). - Tetrahedral bond angle: arccos(−1/3) = 109.47° → cos θ = −1/3. - Td point group: order 24, isomorphic to S4 (permutation of 4 vertices). - Root system A3: 12 roots, e.g., (±1, ±1, 0) permutations; Weyl group = S4 = Td. - Tetrahedral interstitial site coordinates: (1/2, 1/2, 1/2) and (1/4, 1/4, 1/4) in the conventional cell. - Phosphorus substitution: P replaces Si at a tetrahedral site, preserving Td symmetry locally (no Jahn–Teller distortion in the neutral state). - Lattice QCD (arXiv:0710.4339): heavy-quark masses via Fermilab action — no direct geometric ratio; irrelevant to the tetrahedral structure. CONNECTION: - Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com