Proposition: Conditional admissible finite part

Suppose:

1. Theorem thm:main applies.

2. The reference on $V$ is proved.

3. The scale-dependent subtraction ${S_{\delta }}^{\rho }$ is authorized by the actual correction rules.

4. The finite part $T^{\rho }$ satisfies the required intermediate microlocal and compatibility conditions.

Then $T^{\rho }$ is an extension of that reference obtained by the authorized subtraction, and its candidate normalized extensions in the supplied ambiguity class are

For a specified linear normalization map $\mathcal{N}$, a normalization

exists exactly when

When a solution exists, it is unique exactly when

Proof. Theorem thm:main constructs the subtracted limit. Proposition prop:support gives agreement on $V$. Authorization and the stated compatibility conditions supply the additional admissibility requirements.

The normalization equation is the linear equation $\mathcal{N}(a)=-\mathcal{N}(T^{\rho })$. Its solvability and uniqueness are precisely the displayed image and kernel conditions.