Uniform irreducibility and Galois groups of generalized Laguerre polynomials

We prove that the generalized Laguerre polynomial $H_{n,s}$ is irreducible over $\mathbb Q$ and has Galois group containing $A_n$ for every positive integer $s$, once $n$ exceeds an effective absolute constant. Equivalently, every numerator and denominator in the exponential Padé table has these properties once its own degree is sufficiently large. The exceptional positive integral pairs form an effectively computable finite set. The proof combines integral traces at degenerate fibers, Padé resultants, and an energy inequality for Galois-invariant positive semidefinite forms on the roots.