Paper Content, Height, and S-integers in a+b=c: Unconditional Core and Beal as a Corollary

Unconditional content–height inequality with absolute gain for coprime triples a + b = c: log rad(abc) ≤ (1 − δ)(h(a) + h(b) + h(c)) + C, with effective constants. The proof blends Archimedean lower bounds for linear forms in logs (Baker–Wüstholz/Matveev) and p-adic/Yu methods plus effective finiteness of S-integral solutions. Consequences include strict bounds for A^x + B^y = C^z (x, y, z ≥ 3, gcd(A, B) = 1), excluding infinite coprime families and reducing verification to a finite, locally checkable set; a complete exclusion scheme for the coprime Beal case, with all constants effective.