Smooth profinite groups, III: the Smoothness Theorem

Let $p$ be a prime. In this article, we prove the Smoothness Theorem, which asserts that a $(1,1)$-cyclotomic pair is $(n,1)$-cyclotomic, for all $n \geq 1$. In the particular case of Galois cohomology, the Smoothness Theorem provides a new proof of the Norm Residue Isomorphism Theorem, entirely disjoint from motivic cohomology. A byproduct of this approach, is that the latter Theorem follows from mod $p^2$ Kummer theory for fields alone. We moreover extend it, from absolute Galois groups of fields, to algebraic fundamental groups of (not necessarily smooth, nor proper) curves over algebraically closed fields.