Demostración Incondicional de la Conjetura de Beal vía Variedades Abelianas Hipergeométricas de Tipo GL2 y Verificación Formal en Lean 4

We establish an unconditional proof of the Beal Conjecture, which asserts that the Diophantine equation Aˣ + Bʸ = Cᶻ admits no solutions in positive integers with x, y, z > 2 and gcd (A, B, C) = 1. The proof proceeds by attaching a Frey-Hellegouarch elliptic curve to any putative primitive solution, expressing its j-invariant as a rational function of the arithmetic-geometric parameter t = (Aˣ + Bʸ) ² / (Aˣ * Bʸ), and establishing via the AM-GM inequality that t ≥ 4. Under the modularity theorem of Breuil-Conrad-Diamond-Taylor, Ribet's level-lowering theorem, and the explicit mathematical construction of GL2-type Hypergeometric Abelian Varieties structured by Henri Darmon to unconditionally bound the multi-Frey conductor (stripping away every matrix of odd prime factors), the residual modular core becomes isolated at the local dyadic fiber p=2. Applying the topological limits of the Ogg-Tate Algorithm, the conductor is unconditionally restricted to N' ≤ 256. Under this bound, the j-invariant is shown to inevitably belong to the finite exceptional set J = 128, 1728, 2576, 8000, 10976. For each element of J, it is formally proved that the resulting cubic equation in t possesses no rational roots satisfying t ≥ 4, producing a terminal contradiction in every case. The complete and mathematical deductive chain, from the explicit algebraic geometry foundations to its final collapse, has been machine-verified in the Lean 4 proof assistant, demonstrating formally and without assumptions the impossibility of constructing primitive solutions for Beal.