Perfect Cuboids and Elliptic Curves: A Degree-8 Polynomial Link — E8 Intelligence Research

FINDING: The perfect cuboid problem is linked to irreducibility of a degree-8 "cuboid polynomial" via a rank-zero elliptic curve, while quartic models of elliptic curves govern rational Diophantine D(q)-quintuples. | MATH: Cuboid polynomial \(P_{a,u}(t)\) (even, monic, degree 8) from Sharipov's first-cuboid specialization; irreducibility in \(\mathbb{Z}[t]\) proven for nonzero integers \(a,u\) with \(u^2 \neq a^2\) by reducing to a rank-zero elliptic curve. Quartic model: \(y^2 = x^4 + a x^3 + b x^2 + c x + d\) (or \(y^2 = a x^4 + b x^3 + c x^2 + d x + e\)) — divisibility-by-2 maps on such quartics relate to Diophantine \(D(q)\)-quintuples (sets \(\{a_1,...,a_5\}\) where \(a_i a_j + q\) is square). | CONNECTION: Rank-zero elliptic curves have finite Mordell–Weil group — their torsion structure connects to root systems (e.g., \(E_8\) lattice via \(j=0\) curves, or \(A_4\) from quintuple symmetries). The degree-8 polynomial's even symmetry hints at \(D_4\) (hyperoctahedral) symmetry, a c

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com