The 3–5 Switch: Wiles–Taylor Modularity and the Proof of Fermat's Last Theorem — E8 Intelligence Research

FINDING: Wiles–Taylor proved modularity for semistable elliptic curves, forcing FLT via Frey–Serre–Ribet; the "3–5 switch" is the deformation-theoretic pivot that closes the argument. | MATH: Frey curve E: y² = x(x−aᵖ)(x+bᵖ), discriminant Δ = (abc)²ᵖ/2⁸, conductor N = rad(abc) (squarefree, semistable); Ribet: modular ρ̄_{E,p} ⟹ E modular; Wiles: R = T for minimal deformation ring R and Hecke algebra T, with Taylor–Wiles patching over auxiliary primes Q ≡ 1 mod pⁿ; 3–5 switch: residual rep ρ̄₃ (GL₂(𝔽₃), solvable image) ⟹ modularity, then transfer to ρ̄₅ (GL₂(𝔽₅), irreducible) via compatible deformations. | CONNECTION: Modular forms live on the upper half-plane with SL₂(ℤ) action; weight-2 forms ↔ H¹ of modular curve X₀(N); Hecke eigenvalues aₚ satisfy |aₚ| ≤ 2√p (Ramanujan–Petersson, i.e. the 2cos θ bound tied to SU(2) and the Sato–Tate measure (2/π)sin²θ dθ). The 3–5 switch exploits GL₂(𝔽₃) ≅ binary tetrahedral group (order 24, ≅ SL₂(𝔽₃)) — a crystallographic/root-system symmetry (A₃ ≅ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com