Let $(M^n,g)$ be a complete Riemannian manifold with $\operatorname{Ric}_g \ge 0$. Let $u:M\times(0,T]\to(0,\infty)$ be a positive heat solution, and set $f=\log u$. Assume either that $M$ is compact, or that on each time slab $M\times[\varepsilon,T]$ with $\varepsilon>0$ the functions $|\nabla f|$ and $|\partial_t f|$ are bounded and the usual cutoff maximum-principle argument has vanishing error at infinity. Then

\begin{align*} |\nabla f|^2 - \partial_t f \le \frac{n}{2t} \end{align*}

for all $t\in(0,T]$.