\begin{align*}
R: M \times (0,T] &\to \mathbb{R}
\end{align*}
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denote the scalar curvature of $g(\tau)$. Assume that the flow satisfies Hamilton's trace Harnack inequality: for every $\tau \in (0,T]$, every $x \in M$, and every vector $V \in T_xM$,