The Frey Curve: The Bridge Between Fermat's Last Theorem and Modularity — E8 Intelligence Research

FINDING: The Frey curve construction — a semistable elliptic curve with discriminant linked to a hypothetical Fermat counterexample — is the pivotal bridge forcing the contradiction between modularity and level lowering, completing the FLT proof. | MATH: Frey curve: \( E_{a,b,c}: y^2 = x(x-a^p)(x+b^p) \) for \( a^p + b^p = c^p \), \( p \ge 5 \) prime. Discriminant \( \Delta = (abc)^{2p} \cdot 16 \) (up to sign), conductor \( N = \text{rad}(abc) \) (squarefree — semistable). Ribet's level lowering: if \( E \) is modular of level \( N \), then a newform of level \( 2 \) exists (since \( p \mid \Delta \) with \( p \ge 5 \) and \( p \nmid N \)), but no elliptic curves over \( \mathbb{Q} \) have conductor \( 2 \) (or any level \( < 11 \)). Modularity (Taniyama–Shimura–Weil, proved by Wiles–Taylor–Breuil–Conrad–Diamond–Taylor): every semistable elliptic curve over \( \mathbb{Q} \) is modular. Contradiction: \( E \) cannot exist → FLT true. | CONNECTION: The Frey curve's \( j \)-invariant: \( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com