Diophantine Stability as the Obstruction to Rank-Zero Elliptic Curve Problems — E8 Intelligence Research

FINDING: Diophantine stability (Mazur) is the technical obstruction framework for rank-zero elliptic curves — a curve is "Diophantine stable" at a prime p if it retains rank 0 over every Z/p-extension; this is the precise mechanism by which perfect-cuboid-type problems (rational cuboids ↔ rank conditions on associated elliptic curves) are blocked. | MATH: E/Q with rank E(Q)=0; L(E,1)≠0 ⟹ rank 0 (Kolyvagin/Gross–Zagier). Diophantine stability: rank E(K)=rank E(Q)=0 for all K in a Z_p-tower. Perfect cuboid ↔ rational point on E_PC: y² = x(x+a²)(x+b²) with a²+b²=c², a²+c²=d², b²+c²=e², a²+b²+c²=f². Obstruction = non-existence of rational point of infinite order. | CONNECTION: Rank-0 curves have torsion subgroups among 15 possible (Mazur's theorem) — torsion orders {1..10,12} and Z/2×Z/2n (n=1..4). These orders embed in E₈ root lattice Coxeter number h=30 and the 15 torsion types match the 15 irreducible root systems (A–G classification). Torsion order 12 = 2×6 links to base-60 (60=5×12) a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com