The E7 → A3 Canonical Projection Trilogy - Orbit Decomposition and Lattice–Coset Structure of the Canonical A3 Projection of the E7 Root System

A completely analytic, zero-enumeration geometric characterization of the $E_7$ root system under a canonical $A_3$ projection. This three-part series systematically investigates the transition from spherical isotropy to discrete octahedral anisotropy:Part I: Establishes the exact $O_h$-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.