Let $(M^n,g(t))$, $t\in I$, be a smooth Ricci flow, and let $\iota_t:E\to TM$ be a Uhlenbeck moving frame defined on an open spacetime region where the required frame ODE has a smooth solution. Let $\mathcal R(t)$ be the curvature operator pulled back to $\Lambda^2E$ by $\iota_t$. Then $\mathcal R$ satisfies
where $\Delta$ is the rough Laplacian induced by the pulled-back connection and $\mathcal R^2+\mathcal R^{\#}$ is Hamilton's fibrewise quadratic expression in $\mathcal R$ under the curvature-operator normalization used in the proof.