Proposition: Retained-order gauge repair criterion

Let $V^0,V^1,V^2$ be the specified coefficient spaces of a graded complex. Suppose

Let $O_{\mathrm{cl}}\in V^0$ satisfy $s_{\mathrm{cl}}{O}_{\mathrm{cl}}=0$, and let $T\in V^0$ be a constructed order-$\hslash$ coefficient. For a specified linear space of allowed finite corrections $A\subset V^0$, define

Then

and the corrected expression

is closed modulo ${\hslash }^2$ if and only if

Proof. Nilpotence and the anticommutation relation give

The coefficient of $\hslash$ in the corrected closure condition is

Its solvability in the stipulated correction space is exactly (eq:gauge-image).