A completely analytic, zero-enumeration geometric characterization of the E₇ root system under a canonical A₃ projection. This three-part series systematically investigates the transition from spherical isotropy to discrete octahedral anisotropy: Part I: Establishes the exact Oₕ-orbit decomposition and preimage multiplicities (1, 6, 8) using purely diophantine lattice-coset intersections. Part II: Proves the strict delay of spatial anisotropy to the sixth order by demonstrating the exact vanishing of the fourth-order anisotropic coefficient. Part III: Solves the global KKT optimization for the first anisotropic (sixth-order) moment, deriving its exact extrema (21, 24) and the fundamental 7/53 rational anisotropy ratio. All results are strictly derived via invariant and Lie-theoretic methods, without reliance on combinatorial enumeration or numerical sampling.
Ender UYGUN (Fri,) studied this question.