The characterisation of commutativity (paper §3, lemma $commutativity$): a series is commutative if and only if it satisfies the swap and the rotate equations. The forward direction is immediate — the swap and rotate equations are exactly the commutative equivalence of words differing by a transposition or a rotation. The backward direction uses that swaps and rotations connect any two words with the same multiset of letters, so satisfying the two finite families of equations forces the series to be constant on commutatively equivalent words.