A Cyclotomic p-adic Obstruction For a^x + b^y = c^z

This manuscript develops a new “killer prime” obstruction to primitive solutions of the generalized Fermat (Beal) equation a^x+b^y=c^z, with x,y,z≥3. Working in the p-th cyclotomic field, we attach to each rational slope a finite–field obstruction polynomial Φp(t), identified with a hypergeometric trace function for a rigid rank-2 local system. Katz’s equidistribution theorems imply that for any fixed slope there is a density-one set of primes p for which Φp(t)≠0, forcing a λ-adic valuation mismatch and ruling out global solutions for the corresponding exponent patterns.