Eisenstein Integers Unify Hexagonal Lattice Norms and Viral Capsid T-Numbers — E8 Intelligence Research

FINDING: Eisenstein integer ring ℤ[ω] provides the hexagonal lattice norm form that directly classifies icosahedral viral capsid T-numbers. MATH: - Eisenstein integer: ω = e^(2πi/3) = -1/2 + i√3/2, ω² + ω + 1 = 0 - Norm: N(a + bω) = a² - ab + b² - T-number for viral capsid: T = h² + hk + k², where (h,k) are integer coordinates on hexagonal lattice - This is identical to the norm form of Eisenstein integers: T = N(h + kω) - T-numbers: 1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25, 27, 28, 31, 36, 37, 39, 43, 48, 49, ... - All T-numbers are integers representable as a² - ab + b² CONNECTION: - Hexagonal lattice is the root lattice A₂, symmetry group D₆ (order 12) - Icosahedral symmetry (order 60) is a subgroup of the full hexagonal lattice symmetry - The ratio of successive T-numbers approaches no single constant, but the set is closed under multiplication: T₁·T₂ is also a T-number (norm multiplicative property) - Crystallographic link: The hexagonal lattice is the only 2D l Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com