Smooth profinite groups, III: the Smoothness Theorem

Let $p$ be a prime. The goal of this paper is to prove the Smoothness Theorem 5.1, and to explain how it provides a new proof of the Norm Residue Isomorphism Theorem. It states, in particular, that a $(1,\infty)$-cyclotomic pair is $(n,1)$-cyclotomic for all $n \geq 1$, solving by the affirmative part of the Smoothness Conjecture 14.25 of [DCF0], in depth $e=1$.