EFT asymptotics: the growth of operator degeneracy

We establish formulae for the asymptotic growth (with respect to scaling dimension) of number operators in effective field theory, or equivalently $S$-matrix elements, arbitrary spacetime dimensions and with generic content. This we achieve by generalising a theorem due Meinardus applying it Hilbert series -- partition functions degeneracy (subsets of) operators. Although our are asymptotic, numerical experiments reveal remarkable agreement exact results at very low orders EFT expansion, including complicated phenomenological theories such as standard model EFT. Our methods also phase transition-like behaviour series. discuss prospects tightening bounds providing rigorous errors operator degeneracy, extending analytic study utility