Theorem: Conditional fixed-background-input addition

Suppose, on one explicitly specified common carrier and ordering presentation:

1. the inherited tree expression has its stated equation-ideal identity;

2. an admitted one-loop coefficient $\mathbf{\varphi }(\ell )$, with $\ell \in {C_c}^{\infty }(\hat{M})$, has actually been constructed;

3. $\ell$, including its finite insertion and normalization terms, is independent of $h,k$;

4. no untracked ordering or background-transport correction is present at the compared order.

Then

has the same fixed-free-input background identity through that order.

Proof. The tree difference belongs to its equation ideal by (C1). The additional term has identically zero background difference by (C3). Conditions (C2) and (C4) ensure that these are defined elements of the same comparison problem.