[proofplan]
The argument is a direct reduction to Perelman's finite-time extinction theorem for closed three-manifolds with finite fundamental group. Simple connectedness is used only to identify the fundamental group as the trivial group, hence finite. The remaining hypotheses in the statement are exactly the analytic hypotheses under which Perelman's Ricci-flow-with-surgery extinction theorem applies, so applying that theorem gives a finite extinction time for the surgery flow starting from $(M,g_0)$.
[/proofplan]
[step:Reduce simple connectedness to finite fundamental group]
Choose a base point $p \in M$, and let $\pi_1(M,p)$ denote the fundamental group of $M$ at $p$. Since $M$ is simply connected, every loop based at $p$ is homotopic rel endpoints to the constant loop at $p$. Therefore
\begin{align*}
\pi_1(M,p) = \{e\},
\end{align*}
where $e$ denotes the identity element represented by the constant loop. Hence $\pi_1(M,p)$ is finite.
[/step]
[step:Match the surgery flow to Perelman's extinction theorem]
Let $(\mathcal{M}(t))_{t \geq 0}$ denote the Ricci flow with surgery starting from $(M,g_0)$ in the theorem statement, where $\mathcal{M}(0)=M$. Perelman's finite-time extinction theorem for Ricci flow with surgery states the following: if $N$ is a closed three-manifold with finite fundamental group, $h_0$ is a Riemannian metric on $N$, and the surgery flow from $(N,h_0)$ is constructed with Perelman's surgery parameters, canonical-neighbourhood assumptions, Hamilton-Ivey pinching estimates, and noncollapsing controls, then there exists a time $T < \infty$ such that the post-surgery time-slice is empty for every $t \geq T$.
The manifold $M$ is closed by hypothesis, and the preceding step proves that its fundamental group is finite. The initial tensor $g_0$ is a Riemannian metric on $M$ by hypothesis. The flow $(\mathcal{M}(t))_{t \geq 0}$ is assumed in the theorem statement to be constructed with Perelman's surgery parameters, canonical-neighbourhood assumptions, pinching estimates, and noncollapsing controls. Thus every hypothesis of Perelman's finite-time extinction theorem is satisfied with $N=M$ and $h_0=g_0$.
[guided]
We now check the input conditions of the extinction theorem one by one, because the conclusion is not a formal property of arbitrary Ricci flows with surgery. Let $(\mathcal{M}(t))_{t \geq 0}$ be the surgery flow starting from $(M,g_0)$, so $\mathcal{M}(0)=M$. Perelman's finite-time extinction theorem applies to a closed three-manifold $N$ with finite fundamental group, equipped with an initial Riemannian metric $h_0$, provided the flow is constructed with Perelman's surgery parameters, canonical-neighbourhood assumptions, Hamilton-Ivey pinching estimates, and noncollapsing controls.
We take $N=M$ and $h_0=g_0$. The hypothesis says that $M$ is closed, so the closedness condition is met. The previous step showed that, after choosing a base point $p \in M$, the group $\pi_1(M,p)$ is the trivial group $\{e\}$; in particular it is finite. The hypothesis also says that $g_0$ is a Riemannian metric on $M$, so the initial-data condition is met. Finally, the theorem statement specifies that the surgery flow is constructed with Perelman's surgery parameters, canonical-neighbourhood assumptions, pinching estimates, and noncollapsing controls. These are precisely the surgery and analytic assumptions required by the extinction theorem. Therefore Perelman's finite-time extinction theorem applies to $(\mathcal{M}(t))_{t \geq 0}$.
[/guided]
[/step]
[step:Conclude extinction in finite time]
By the application in the preceding step, there exists a real number $T < \infty$ such that
\begin{align*}
\mathcal{M}(t)=\varnothing \qquad \text{for every } t \geq T.
\end{align*}
This is exactly the assertion that the Ricci flow with surgery starting from $(M,g_0)$ becomes extinct in finite time.
[/step]
