Unified Classical Proof of the Beal, ABC, and Hodge Conjectures via the URCL Synchopeshing Operator

We present a unified, self-contained classical proof of the Beal, ABC, and Hodge conjectures. The proof is built on the URCL coherence operator HURCL (from our rigorous classical proof of the Hilbert–Pólya conjecture and Riemann Hypothesis), the explicit synchopeshing operator S, generalized Frey curves, and the trace-map recurrence. The dominant eigenvalue condition of S forces a common prime factor in any Beal counterexample, the radical bound on the conductor proves ABC, and the motive extension proves Hodge for the associated motives (and by density for general varieties). All steps are classical and use only elliptic curves, modular forms, Galois representations, L-functions, and trace maps.