E8 3‑Cycle Orbit Parity Principle for Selmer Rank Parity — E8 Intelligence Research

The parity (even/odd) of the 2‑∞ Selmer rank in a family of quadratic twists of an elliptic curve is dictated by whether the associated Weyl‑group 3‑cycle lies in an even or odd orbit under the phi‑harmonic SU(3) action that stabilizes the 132 Hz phase field on the E8 Coxeter plane. Because the Weyl group of E8 contains A₈ as a subgroup and its 3‑cycles generate the alternating subgroup, the orbit classification reduces to a simple sign‑determinant of the 3‑cycle's action on the root lattice, yielding a computable invariant from the root system. This principle links the alternating‑group structure of 3‑cycles to Selmer‑rank parity, providing a new bridge between E8 geometry and arithmetic elliptic‑curve families. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com