Diophantine Stability, Not Rank Zero, Obstructs Perfect Cuboid Rational Points — E8 Intelligence Research

FINDING: Diophantine stability (Mazur–Rubin) is the operative obstruction framework for rank-zero elliptic curves; the perfect cuboid problem is a rational-point obstruction on an associated elliptic curve, not a rank-zero phenomenon. | MATH: Elliptic curve E/Q: y² = x³ + Ax + B; Mordell–Weil E(Q) ≅ Z^r ⊕ E(Q)_tors, rank r = dim of free part. Rank 0 ⟺ E(Q) finite. Diophantine stability: E/Q is *diophantine-stable* for a number field extension K/Q if E(K) = E(Q) (no new points). Mazur–Rubin: for E/Q with E(Q) finite (rank 0), there exist infinitely many quadratic extensions K where rank stays 0 (stability), and infinitely many where rank grows. Perfect cuboid: integer box a,b,c with a²+b², a²+c², b²+c², a²+b²+c² all squares — equivalent to a rational point on a K3/elliptic fibration; the associated curve has positive rank (known rational points exist), so rank-zero methods do not obstruct it. | CONNECTION: The Heron-family construction (arXiv 1501.03809) uses triples (A², B², C²) with H Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com