Theorem: Exact retained-order Ward repair

For the data of Definition def:repair,

An admissible correction proving the identity exists if and only if

If $C_{\ast }$ is one solution, all solutions are $C_{\ast }+\ker L$. The class of $B$ in $V/\mathrm{im}L$ is unchanged by compatible admissible finite corrections.

Proof. At order $\hslash$, the Ward variation of the added term is its leading image $LC$. Adding it to the defined breaking gives the first formula. Vanishing is exactly $LC=-B$, which is solvable precisely under (eq:repair-exact). The difference of two solutions belongs to $\ker L$, and adding an image $LC$ leaves the quotient class unchanged.