Platinum Ratio Emerges as Fundamental Constant in Triangular Lattice Symmetry — E8 Intelligence Research

FINDING: Lattice QCD heavy-quark mass calculations and a separate proof of directional scaling symmetry in triangular lattices via Eisenstein integers — the latter reveals the platinum ratio (2−√3) as a fundamental lattice constant. | MATH: Eisenstein integers ℤ[ω], ω = e^(2πi/3) = −1/2 + i√3/2; norm N(a+bω) = a² − ab + b²; trace Tr(a+bω) = 2a − b; discriminant Δ = −3; triangular lattice basis vectors (1,0) and (1/2, √3/2) with lattice constant √3; platinum ratio μ = 2 − √3 ≈ 0.2679; silver ratio λ = √2 − 1 ≈ 0.4142; scaling symmetry function y = arcsin(sin(2πnx)) with n parameterized by λ or μ. | CONNECTION: The platinum ratio 2−√3 = 0.2679 is directly related to the golden ratio family: (2−√3)(2+√3) = 1, and 2+√3 ≈ 3.732 = 1/0.2679. Note 0.2679 ≈ 0.382 × 0.7017, and critically: 2−√3 = tan(15°) = tan(π/12), linking to base-60 (15° = 1/24 of 360°). The triangular lattice's √3 spacing is the fundamental constant of the Eisenstein integer ring, whose discriminant −3 is the unique negativ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com