The Beal Conjecture via Frey Curves: Partial Results, a Corrected Identity, and an Obstruction Analysis

An earlier version of this work claimed a complete proof of the Beal conjecture by the modular method. That claim was incorrect and is withdrawn; the Notice of Correction below identifies the errors. The present version retains and develops what is sound: a corrected and strengthened form of the Palindromic Identity for the j-invariant of the Frey curve, with the existence conditions required for its valid use; an irreducibility theorem for the mod-q Galois representations of Beal Frey curves for all primes q ≥ 7, proved by a new route through rational 2-torsion and the Mazur–Kenku classification, with the case q = 5 established computationally; a corrected proof that the conductor exponent at 2 satisfies f₂ ≤ 4; and an obstruction theorem showing that no choice of auxiliary prime can make the single-prime elliptic Frey curve method prove the general mixed-exponent case. The Beal conjecture remains open.