[proofplan] We prove the conditional surgery-existence statement by using the assumed Perelman surgery package as the continuation mechanism. First we choose the surgery hierarchy $\delta,h,D$ supplied by the package for the fixed data $(g_0,T,\varepsilon)$. We then run the smooth Ricci flow until either extinction or the trigger curvature $D h^{-2}$ is reached, perform surgery on the canonical strong $\delta$-necks, and use the cap and persistence estimates to restart the flow with the same pinching and noncollapsing hypotheses. The finite-time nonaccumulation theorem prevents infinitely many surgeries before $T$, so the iterative construction reaches $T$ unless the manifold becomes extinct earlier. [/proofplan] [step:Choose the surgery parameter hierarchy for the fixed initial data] Let $M$ be the given closed oriented three-manifold and let $g_0$ be the given smooth Riemannian metric on $M$. The orientation is part of the input class for the assumed Perelman surgery package; the construction below does not otherwise use orientation except to keep all surgery cuts and cap insertions within that oriented category. Fix the finite time horizon $T<\infty$ and the accuracy parameter $\varepsilon>0$. By the assumed Perelman surgery package applied to the data $(M,g_0,T,\varepsilon)$, there is a number $\delta_0(g_0,T,\varepsilon)>0$ such that, for every $\delta$ with $0<\delta\leq \delta_0(g_0,T,\varepsilon)$, the package provides positive constants $h=h(g_0,T,\varepsilon,\delta)$ and $D=D(g_0,T,\varepsilon,\delta)$, where $\delta$ is used as the constant cutoff function $\delta(t)\equiv\delta$ on $[0,T]$, where $h$ is the surgery neck radius, and where $D h^{-2}$ is the trigger scalar-curvature scale. These constants are chosen in the prescribed hierarchy: first $\delta$, then $h$, then $D$, and with $D h^{-2}>\sup_{x\in M}R_{g_0}(x)$, where $R_{g_0}:M\to\mathbb{R}$ is the scalar curvature of $g_0$. The assumed package includes the canonical-neighbourhood theorem, the cutoff and surgery-selection theorem that chooses disjoint strong $\delta$-necks, the standard-cap insertion theorem, the post-surgery persistence estimates, and the finite-time nonaccumulation theorem. Thus every subsequent canonical-neighbourhood, surgery-selection, cap-insertion, persistence, and noncollapsing estimate is valid on the interval $[0,T]$, and the initial slice begins below the trigger scale. [/step] [step:Run the smooth flow until extinction or the trigger scale] Let \begin{align*} g: M \times [0,\tau) &\to S^2T^*M \end{align*} be the maximal smooth Ricci flow with initial value $g(0)=g_0$, where $S^2T^*M$ denotes the bundle of symmetric covariant two-tensors on $M$, and where $g$ satisfies \begin{align*} \partial_t g(t)=-2\operatorname{Ric}_{g(t)}. \end{align*} The short-time existence theorem for smooth Ricci flow on closed manifolds gives such a solution on a nonempty interval, and closedness of $M$ verifies the compactness hypothesis needed for the standard smooth existence theorem. Since the hierarchy was chosen so that $D h^{-2}>\sup_{x\in M}R_{g_0}(x)$, the trigger has not occurred at time $0$. Continue the smooth solution until the first time $t_1\leq T$ at which \begin{align*} \sup_{x\in M(t_1)} R(x,t_1)=D h^{-2}, \end{align*} where $M(t_1)$ is the time-$t_1$ manifold and $R(\cdot,t_1):M(t_1)\to\mathbb{R}$ is the scalar curvature of $g(t_1)$. If no such trigger time occurs before $T$, the smooth Ricci flow itself is the desired surgery flow on $[0,T]$ with no surgeries. Extinction is therefore not a stopping alternative for this initial smooth interval; it will arise only after later surgery and component discarding. [guided] The first task is to make the continuation rule precise. We let \begin{align*} g: M \times [0,\tau) &\to S^2T^*M \end{align*} be the smooth Ricci flow starting from $g_0$, meaning that $g(0)=g_0$ and \begin{align*} \partial_t g(t)=-2\operatorname{Ric}_{g(t)}. \end{align*} The smooth short-time existence theorem applies because the initial manifold is closed and $g_0$ is smooth. The scalar curvature at time $t$ is the function \begin{align*} R(\cdot,t):M(t)&\to\mathbb{R}, \end{align*} where $M(t)$ denotes the current time slice. At time $0$, this scalar curvature is $R_{g_0}:M\to\mathbb{R}$, and the surgery hierarchy has chosen $D$ so that \begin{align*} \sup_{x\in M}R_{g_0}(x)<D h^{-2}. \end{align*} Thus the initial metric is not already past the cutoff scale. We stop the initial smooth flow at the first time $t_1\leq T$ when the largest scalar curvature reaches the prescribed trigger value: \begin{align*} \sup_{x\in M(t_1)} R(x,t_1)=D h^{-2}. \end{align*} If this trigger time does not occur before $T$, then no surgery is needed on $[0,T]$, and the smooth Ricci flow is already the required flow with surgery in the degenerate sense of having no surgery times. This distinction matters because the initial smooth flow is defined on the fixed manifold $M$; extinction is not a literal event during this first smooth interval. Extinction becomes possible only after surgery, when the cutoff construction may discard all remaining components. The parameter $D h^{-2}$ is exactly the scale at which the surgery package is invoked: below this scale the flow is left untouched, while at this scale the canonical-neighbourhood and surgery-selection theorems from the assumed package identify disjoint strong $\delta$-necks in the high-curvature region. [/guided] [/step] [step:Cut along canonical strong $\delta$-necks and insert standard caps] Assume the first stopping time $t_1$ is not an extinction time. The canonical-neighbourhood theorem in the assumed surgery package applies at the trigger scale $D h^{-2}$ because $D$ was chosen after $h$ and $\delta$ in the hierarchy. The cutoff and surgery-selection theorem within the same package then upgrades the local canonical-neighbourhood information to a locally finite collection of pairwise disjoint strong $\delta$-necks with scalar curvature comparable to $h^{-2}$ that separate the components to be discarded from the components to be continued. Perform surgery along this disjoint neck collection. On each retained boundary component insert a copy of the fixed standard cap, rescaled by the factor $h^2$, so that after scaling by $h^{-2}$ the inserted cap is $\delta$-close to the standard cap model in the topology required by the package. Denote the post-surgery manifold immediately after time $t_1$ by $M_+(t_1)$ and the post-surgery metric by $g_+(t_1)$. [/step] [step:Restart the flow using the persistence estimates] The cap construction and persistence part of the assumed surgery package state that $g_+(t_1)$ satisfies the same [Hamilton-Ivey pinching estimate](/theorems/6022), the required curvature bounds at the cutoff scale, and the noncollapsing constants needed to restart the Ricci flow. Therefore the smooth short-time existence theorem applies to the post-surgery initial data $(M_+(t_1),g_+(t_1))$, and we obtain a smooth Ricci flow on a next interval $[t_1,t_2)$. Repeating the same stopping rule, we either reach time $T$, discard all remaining components after a later surgery and hence become extinct, or perform another surgery at a later trigger time. [guided] The restart step is where the surgery package must restore the hypotheses needed for the next smooth interval. After cutting along the selected strong $\delta$-necks and inserting standard caps, the post-surgery initial data are the closed three-manifold $M_+(t_1)$ and the smooth metric $g_+(t_1)$. The cap construction and persistence estimates in the assumed package assert that this metric satisfies the Hamilton-Ivey pinching estimate, the curvature bounds at the cutoff scale, and the noncollapsing constants required by the same package. These are exactly the hypotheses that must be available again before the construction can be iterated. Because $M_+(t_1)$ is closed and $g_+(t_1)$ is smooth, the smooth short-time existence theorem for Ricci flow applies to the initial data $(M_+(t_1),g_+(t_1))$. Thus there is a smooth Ricci flow beginning at time $t_1$ and defined on a nonempty interval $[t_1,t_2)$. On this next interval we use the same trigger rule: continue the smooth flow until either time $T$ is reached or the scalar curvature reaches $D h^{-2}$ again. If the later surgery-selection and discarding step removes every component, then the flow is extinct; otherwise the same cap insertion and persistence argument restarts the construction again. [/guided] [/step] [step:Use finite-time nonaccumulation to complete the construction on $[0,T]$] Let $(t_j)_{j\in J}$ be the strictly increasing sequence of surgery times produced by the iteration, where $J\subset\mathbb{N}$ is an initial interval of indices. The finite-time nonaccumulation theorem in the assumed surgery package applies because every surgery time was chosen using the fixed hierarchy $\delta,h,D$, every surgery neck is a strong $\delta$-neck at scalar curvature comparable to $h^{-2}$, and the post-surgery pinching, cutoff-scale curvature bounds, and noncollapsing constants are restored after each surgery. Hence the set $\{t_j\in[0,T]:j\in J\}$ is finite. It follows that the iterative construction cannot accumulate before $T$; therefore it either reaches time $T$ after finitely many surgeries or becomes extinct at some time $t_{\mathrm{ext}}\leq T$. This gives Perelman's $\delta(t)$-cutoff Ricci flow with surgery from $(M,g_0)$ on $[0,T]$ unless extinction occurs earlier, with precisely the neck, cap, pinching, cutoff-scale, and noncollapsing properties stated in the theorem. [/step]