Lemma: Protection of lower coefficients

Let two action or operator data sets differ by ${\kappa }^{N}V$, with $N\ge 1$. An expression obtained by finitely many additions, multiplications, coefficientwise linear maps, derivatives, and formal inversions with invertible order-zero part has the same coefficients below order $N$, provided the construction uses no negative coupling powers.

Proof. All listed operations preserve the filtration ${\kappa }^j$. For a formal inverse, the coefficient recursion starts with the common order-zero inverse and determines its $j$-th coefficient from data of orders at most $j$. Induction proves the assertion.