Let $M$ be a smooth $n$-dimensional manifold, let $T \in (0,\infty]$, and let $g: M \times [0,T) \to S^2T^*M$ be a smooth Ricci flow, so that

\begin{align*} \partial_t g(t) = -2\operatorname{Ric}_{g(t)} \end{align*}

for every $t \in [0,T)$. Let $\operatorname{Rm}(t)$ denote the Riemann curvature tensor of $g(t)$, let $\nabla$ denote the Levi-Civita connection of $g(t)$, and let $\Delta_{g(t)}$ denote the scalar Laplace-Beltrami operator acting on functions. Then there exists a constant $C_n > 0$, depending only on $n$, such that the pointwise inequality

\begin{align*} (\partial_t - \Delta_{g(t)})|\operatorname{Rm}(t)|_{g(t)}^2 \leq -2|\nabla \operatorname{Rm}(t)|_{g(t)}^2 + C_n|\operatorname{Rm}(t)|_{g(t)}^3 \end{align*}

holds on $M \times (0,T)$.