The Frey Curve Bridge: Linking Fermat's Last Theorem to Modularity — E8 Intelligence Research

FINDING: The Frey curve construction links Fermat's Last Theorem to modularity via Galois representations, with the discriminant Δ = (abc)²ᵖ forcing a semistable elliptic curve whose 3-adic and p-adic Galois images are constrained by Serre's conjecture. | MATH: Frey curve: y² = x(x − aᵖ)(x + bᵖ); discriminant Δ = (abc)²ᵖ / 256; conductor N = rad(abc) (squarefree, semistable); Galois representation ρE,p: Gal(ℚ̄/ℚ) → GL₂(𝔽ₚ); modular form weight 2, level N; Ribet's level-lowering: ρE,p ≅ ρf,𝔭 for f of weight 2, level 2 (impossible — no such cusp form exists). | CONNECTION: The Frey curve's discriminant is a perfect 2p-th power — the exponent 2p creates a ratio structure where the j-invariant j(E) = 256(a²ᵖ + b²ᵖ)³/(abc)²ᵖ scales with (abc)⁻²ᵖ, echoing the 0.618/1.618 golden ratio inversions in the sense that the curve's arithmetic complexity (conductor) is minimized (squarefree) while its discriminant is maximally powered — a dual extremality. The semistable re Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com