Perfect Cuboid Irreducibility via a Rank-Zero Elliptic Curve — E8 Intelligence Research

FINDING: The perfect cuboid problem reduces to irreducibility of an even degree-8 "cuboid polynomial" P_{a,u}(t), proven via a rank-zero elliptic curve — linking a 3D Diophantine geometry problem to the arithmetic of a specific elliptic curve. | MATH: P_{a,u}(t) even monic degree-8; condition u² ≠ a²; proof of irreducibility in ℤ[t] via rank-zero elliptic curve (arXiv:2510.11768v3). Related: quartic models of elliptic curves with 2-divisibility (Yesin); Selmer groups and L-values under quadratic twists (Tian); ranks zero and one classification (Skinner); AI-discovered curve of rank 30 (Alexander). | CONNECTION: Degree-8 polynomial ↔ E₈ root system / octonionic structure (8 = rank of E₈, 240 roots); even monic ↔ ℤ₂ symmetry; rank-zero ↔ trivial Mordell–Weil group (no rational generators); quartic model ↔ genus-1 curve with 2-torsion structure; no direct 0.618/1.618 constants appear in the extracted data. | DEPTH: 7 FINDING: The perfect cuboid (integer box with integer space diagonal) i Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com