Modularity Theorem: The Bridge Proving Fermat's Last Theorem — E8 Intelligence Research

FINDING: The modularity theorem (Taniyama–Shimura–Weil) is the central bridge linking elliptic curves to modular forms, and its violation would imply Fermat's Last Theorem fails — but no such violation exists, so FLT holds. | MATH: For a semistable elliptic curve \(E: y^2 = x^3 + ax + b\) with discriminant \(Δ_E\), the Frey curve associated to a hypothetical FLT solution \(a^p + b^p = c^p\) has \(Δ_E = 2⁻⁸(abc)²ᵖ\), a prime-power structure (up to sign) that forces \(Δ_E\) to be a perfect \(p\)-th power times a small constant. The modularity theorem states every rational elliptic curve is modular: \(L(E,s) = L(f,s)\) for a weight-2 newform \(f\). Wiles proved this for semistable curves, directly killing the Frey curve's existence. | CONNECTION: The discriminant's prime-power form \(2⁻⁸(abc)²ᵖ\) is a stark violation of the "minimal discriminant" lattice structure — the exponent \(2p\) is not of the form \(12k\) (as required for a cuspidal eigenform of weight 2) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com