[proofplan] We construct the flow by the standard Perelman parameter induction: run the smooth Ricci flow until the first curvature threshold is reached, perform surgery on the necks supplied by the canonical-neighbourhood theorem, attach standard caps at the prescribed scale, and restart the smooth flow. The choice of decreasing surgery parameters is made on successive finite time intervals so that Hamilton-Ivey pinching, canonical neighbourhoods, and noncollapsing are preserved after every restart. The no-accumulation theorem for admissible surgeries gives finitely many surgery times on each compact time interval; hence the iterative construction either becomes extinct or continues for all positive time. [/proofplan]

[step:Record the finite-time surgery theorem and the surgery vocabulary used below] We use the following finite-time Hamilton-Perelman surgery theorem as the external existence-and-continuation input. For every closed orientable three-dimensional Ricci-flow initial metric and every already-constructed admissible surgery history on a compact interval, the theorem supplies positive parameters $r$, $\kappa$, $\delta_{\mathrm{max}}$, $h_{\mathrm{max}}$, and a curvature cutoff $\Theta$ with this property: if each new surgery in the next compact time slab is a $\delta$-strong-neck excision at radius $h$, with $0<\delta\leq\delta_{\mathrm{max}}$ and $0<h\leq h_{\mathrm{max}}$, then the smooth flow can be continued until the next cutoff time, the high-curvature cutoff region is exactly decomposed into the strong necks, standard-cap regions, and extinct positively curved components specified by the surgery theorem, and the post-surgery metric is again an admissible input for the next continuation step. The same theorem gives Hamilton-Ivey pinching, the canonical-neighbourhood assumption at scale $r$, Perelman noncollapsing with constant $\kappa$ on controlled balls, standard-cap estimates after surgery, and no accumulation of admissible surgery times on the compact interval. Here an admissible surgery history means a finite concatenation of smooth Ricci-flow segments on closed orientable three-manifolds, joined only by the preceding theorem's $\delta$-strong-neck excisions and standard-cap attachments, with components discarded exactly when the theorem's extinction rule classifies them as extinct. A controlled ball means a geodesic ball in a time-slice metric whose curvature, parabolic neighbourhood size, and radius range satisfy the hypotheses in the finite-time theorem; in the present construction those radii are bounded below by a fixed multiple of the active surgery radius and bounded above by the active canonical or macroscopic scale. The cutoff criterion is that the current smooth segment is stopped at the first time when the curvature threshold $\Theta$ prescribed for the active slab is reached. [/step]

[step:Start the maximal smooth Ricci flow from the closed initial metric] Let $M_0 := M$. Since $M$ is closed, smooth, orientable, and three-dimensional, $M_0$ is a compact smooth orientable three-manifold without boundary. Let $\mathcal{S}_2(M_0)$ denote the bundle of symmetric covariant two-tensors on $M_0$. Let \begin{align*} g: M_0 \times [0,T_1) &\to \mathcal{S}_2(M_0) \end{align*} be the maximal smooth Ricci flow with initial value $g(0)=g_0$, meaning $g(x,t)=g(t)_x$ and $\partial_t g(t)=-2\operatorname{Ric}_{g(t)}$ for all $t\in[0,T_1)$. The closedness of $M_0$ supplies compactness, so the short-time existence and maximal-extension theorem for Ricci flow on closed manifolds applies to the smooth Riemannian metric $g_0$. If $T_1=\infty$ and no surgery curvature threshold is ever reached, this smooth flow is already the required Ricci flow with surgery with empty surgery set. [/step]

[step:Choose admissible surgery parameters by compact-time induction]We choose the parameters inductively over the integer intervals. Suppose that the surgery flow has already been constructed on $[0,j-1]$, where for $j=1$ this means only the initial metric $g_0$ is given. Apply the finite-time Hamilton-Perelman surgery theorem recorded above to the existing compact-time history on $[0,j-1]$ and the target interval $[0,j]$. Its hypotheses are satisfied because the initial manifold is closed, orientable, and three-dimensional, every already-constructed piece is a smooth Ricci-flow segment, and every surgery already performed belongs to the admissible history in the sense defined above. The theorem supplies constants $r_j=r_j(g_0,j,\mathcal{H}_{j-1})>0$, $\kappa_j=\kappa_j(g_0,j,\mathcal{H}_{j-1})>0$, $\delta_{j,\mathrm{max}}=\delta_{j,\mathrm{max}}(g_0,j,\mathcal{H}_{j-1})>0$, and $h_{j,\mathrm{max}}=h_{j,\mathrm{max}}(g_0,j,\mathcal{H}_{j-1})\in(0,r_j)$, where $\mathcal{H}_{j-1}$ denotes the already-constructed surgery history up to time $j-1$. Its conclusion is that, if all surgeries performed during the new slab $[j-1,j]$ have neck-quality error at most $\delta_{j,\mathrm{max}}$ and surgery radius at most $h_{j,\mathrm{max}}$, then the resulting flow on $[0,j]$ satisfies the [Hamilton-Ivey pinching estimate](/theorems/6022), the canonical-neighbourhood assumption at scale $r_j$, and Perelman noncollapsing with constant $\kappa_j$ on controlled balls in the permitted radius range. Choose slab parameters $h_j$ and $\delta_j$ satisfying \begin{align*} 0<h_j\leq h_{j,\mathrm{max}}, \qquad 0<\delta_j\leq \delta_{j,\mathrm{max}}. \end{align*} Define the active interval index $J:[0,\infty)\to\mathbb{N}$ by $J(t)=1$ for $0\leq t\leq1$ and $J(t)=j$ for $j-1<t\leq j$. A surgery at time $t$ uses the radius $h_{J(t)}$ and the neck-quality bound $\delta_{J(t)}$. Define the corresponding curvature cutoff function $\Theta:[0,\infty)\to(0,\infty)$ to be constant on each slab, with $\Theta(t)=\Theta_j$ for $j-1<t\leq j$, where $\Theta_j$ is the curvature threshold prescribed by the finite-time surgery induction theorem for the pair $(h_j,\delta_j)$ on $[j-1,j]$. This compact-time induction avoids any requirement that surgeries performed in earlier slabs satisfy bounds chosen only later.[/step]

[guided]The key point is that one cannot choose a decreasing sequence after the fact and then claim that earlier surgeries used the later, smaller radii. Earlier surgeries have already happened. Therefore the correct construction is a compact-time induction: before constructing the slab $[j-1,j]$, we apply the finite-time Hamilton-Perelman surgery induction theorem to the history already constructed on $[0,j-1]$ and ask only for admissible parameters for the new slab. Let $\mathcal{H}_{j-1}$ denote the already-constructed surgery history up to time $j-1$. For $j=1$, $\mathcal{H}_0$ is empty and the only input is the closed orientable three-dimensional initial manifold $(M,g_0)$. The finite-time surgery theorem applies because the existing history consists of smooth Ricci-flow pieces joined by admissible standard surgeries, and the orientability hypothesis ensures that the neck excisions and standard-cap attachments remain in the orientable category. The theorem supplies constants $r_j=r_j(g_0,j,\mathcal{H}_{j-1})>0$, $\kappa_j=\kappa_j(g_0,j,\mathcal{H}_{j-1})>0$, $\delta_{j,\mathrm{max}}=\delta_{j,\mathrm{max}}(g_0,j,\mathcal{H}_{j-1})>0$, and $h_{j,\mathrm{max}}=h_{j,\mathrm{max}}(g_0,j,\mathcal{H}_{j-1})\in(0,r_j)$. Here $r_j$ is the canonical-neighbourhood scale on the compact interval $[0,j]$, $\kappa_j$ is the noncollapsing constant on controlled balls in that interval, $\delta_{j,\mathrm{max}}$ is the permitted neck-quality error for surgeries still to be performed in the slab $[j-1,j]$, and $h_{j,\mathrm{max}}$ is the permitted upper bound for their surgery radius. The dependence on $\mathcal{H}_{j-1}$ is essential: the theorem is being used to extend a fixed past history, not to retroactively impose new restrictions on past surgeries. We now choose the actual slab parameters by \begin{align*} 0<h_j\leq h_{j,\mathrm{max}}, \qquad 0<\delta_j\leq\delta_{j,\mathrm{max}}. \end{align*} Define $J:[0,\infty)\to\mathbb{N}$ by $J(t)=1$ for $0\leq t\leq1$ and $J(t)=j$ for $j-1<t\leq j$. Thus a surgery at time $t$ belongs to the slab indexed by $J(t)$ and uses the radius $h_{J(t)}$ and neck-quality bound $\delta_{J(t)}$. We also define the slabwise curvature cutoff $\Theta(t)=\Theta_j$ for $j-1<t\leq j$, where $\Theta_j$ is the threshold prescribed by the finite-time theorem for $(h_j,\delta_j)$. With these definitions, every surgery in the new slab satisfies exactly the admissibility bounds required for the [extension theorem](/theorems/59) on that slab. No assertion is made that earlier surgeries used the later parameters; their admissibility is part of the already-constructed history.[/guided]

[step:Perform surgery at each curvature threshold and restart the Ricci flow] Assume the flow has been constructed on $[0,t_s)$ after the previous surgery time, and let $j=J(t_s)$ be the active interval index. More precisely, $t_s$ is the first time in the current smooth segment at which the curvature cutoff $\Theta_j$ for the slab $[j-1,j]$ is reached; if no such time occurs before the end of the slab, the smooth segment continues to the next integer time and the parameters for the next slab are chosen by the preceding compact-time induction. If $t_s$ exists, all surgeries already performed in the current slab have radius at most $h_j\leq h_{j,\mathrm{max}}$ and quality at most $\delta_j\leq\delta_{j,\mathrm{max}}$, while all earlier surgeries belong to the fixed admissible history $\mathcal{H}_{j-1}$. Therefore the hypotheses of the finite-time Hamilton-Perelman surgery theorem are satisfied on the active slab. Its full surgery-decomposition conclusion, not merely the canonical-neighbourhood conclusion, applies at the cutoff time: the high-curvature region is decomposed into the strong necks, caps, and positively curved components specified by the theorem. Choose the strong necks selected by that decomposition. Cutting along those necks, discarding precisely the components classified by the extinction rule, and gluing standard caps of radius $h_j$ and quality $\delta_j$ produces a smooth closed orientable three-manifold $M_{s,+}$ and a smooth post-surgery metric $g_{s,+}$. The standard-cap estimates included in the same finite-time theorem give the post-surgery Hamilton-Ivey pinching, canonical-neighbourhood, and controlled noncollapsing assumptions needed for the next smooth segment. Applying short-time existence for Ricci flow on the closed smooth manifold $(M_{s,+},g_{s,+})$ gives the next smooth Ricci-flow segment. [/step]

[step:Preserve pinching, canonical neighbourhoods, and controlled noncollapsing through the compact-time induction] We prove the estimates by induction over the integer slabs. On $[0,1]$, the finite-time Hamilton-Perelman surgery induction theorem applies with the empty initial history and the parameters $(h_1,\delta_1)$, so the smooth pieces and surgeries satisfy the Ricci flow equation, Hamilton-Ivey pinching, canonical-neighbourhood control at scale $r_1$, and Perelman noncollapsing with constant $\kappa_1$ on controlled balls whose radii are at least comparable to $h_1$ and at most the relevant canonical or macroscopic scale. If the assertions hold on $[0,j-1]$, then the constructed history is the admissible input $\mathcal{H}_{j-1}$ for the next application of the finite-time theorem; the choices $h_j\leq h_{j,\mathrm{max}}$ and $\delta_j\leq\delta_{j,\mathrm{max}}$ make every new surgery in $[j-1,j]$ admissible, and the standard-cap estimates restore the required hypotheses immediately after each surgery. Therefore the concatenation of the smooth segments and surgery transitions is a Ricci flow with surgery starting from $g_0$ and satisfying the asserted Hamilton-Ivey, canonical-neighbourhood, and controlled-noncollapsing estimates on every compact interval on which it is defined. [/step]

[step:Exclude accumulation of surgery times before extinction] Let $0<A<\infty$ and let $\mathcal{S}_A\subset [0,A]$ denote the set of surgery times before extinction. Choose an integer $j\geq A$. The compact-time induction has constructed an admissible surgery flow on $[0,j]$ with fixed Hamilton-Ivey pinching control, canonical-neighbourhood scale $r_j$, noncollapsing constant $\kappa_j$, and slabwise surgery parameters $h_i,\delta_i$ for $1\leq i\leq j$, each chosen within the admissible bounds required when its slab was constructed. The no-accumulation conclusion included in the finite-time Hamilton-Perelman surgery theorem applies to this constructed admissible flow on $[0,j]$, and hence to the restricted interval $[0,A]$, giving that $\mathcal{S}_A$ is finite. Since $A$ was arbitrary, there are only finitely many surgery times in each compact time interval before extinction. [/step]

[step:Conclude the extinction alternative] The iterative construction can stop only if the post-surgery manifold is empty, in which case the flow becomes extinct at that surgery time. If this never happens, then the finiteness of surgery times on every compact interval prevents finite-time accumulation, so the construction can be restarted after every surgery and extended past every finite time $A>0$. Thus the Ricci flow with surgery either continues until extinction or exists for all $t\geq 0$, and it has precisely the estimates stated in the theorem. [/step]