[proofplan] We make the thick-thin decomposition using Perelman's long-time theorem in its curvature-scale formulation. For each point $x$ at a nonsurgery time $t$, the relevant scale is the curvature scale $\rho(x,t)$, and the $w$-thick condition is the single volume inequality for the ball $B_{g(t)}(x,\rho(x,t))$. Perelman's theorem supplies constants $w_0>0$ and $T(w)$, uses the surgery-scale hypothesis $o(\sqrt{t})$, gives subsequential smooth hyperbolic limits at thick basepoints after rescaling by $t^{-1}$, and identifies the complement as locally collapsed under a lower sectional-curvature bound. [/proofplan] [step:Define the thick and thin subsets using the curvature scale] Let $(M(t), g(t))$ denote the given three-dimensional Ricci flow with surgery, defined at the nonsurgery time $t$. Define the rescaled Riemannian metric $\tilde g_t$ on $M(t)$ by $\tilde g_t := t^{-1} g(t)$. For $x \in M(t)$, let $\rho(x,t) \in (0,\sqrt{t}]$ denote the curvature scale at $(x,t)$: it is the largest radius $r \leq \sqrt{t}$ such that the sectional curvature of $g(t)$ satisfies $\sec_{g(t)} \geq -r^{-2}$ on the geodesic ball $B_{g(t)}(x,r)$. Say that $x$ is $w$-thick at time $t$ if \begin{align*} \operatorname{Vol}_{g(t)}(B_{g(t)}(x,\rho(x,t))) \geq w\rho(x,t)^3. \end{align*} Let $M_{\mathrm{thick}}(t,w) \subset M(t)$ be the set of $w$-thick points, and define $M_{\mathrm{thin}}(t,w) := M(t) \setminus M_{\mathrm{thick}}(t,w)$. Equivalently, $x \in M_{\mathrm{thin}}(t,w)$ exactly when \begin{align*} \operatorname{Vol}_{g(t)}(B_{g(t)}(x,\rho(x,t))) < w\rho(x,t)^3. \end{align*} These two sets are disjoint and their union is $M(t)$ by construction. [/step] [step:Apply Perelman's long-time theorem to thick basepoint sequences] We invoke Perelman's Long-Time Thick-Thin Theorem for Ricci Flow with Surgery in its curvature-scale form. The theorem provides a constant $w_0>0$ such that, for every $w \in (0,w_0)$, there is a time $T(w)>0$ with the following conclusion for every nonsurgery time $t \geq T(w)$, provided the surgery scale is $o(\sqrt{t})$: any sequence of pointed rescaled manifolds $(M(t_k),t_k^{-1}g(t_k),x_k)$ with $t_k \to \infty$ and $x_k \in M_{\mathrm{thick}}(t_k,w)$ has a subsequence converging smoothly in the pointed Cheeger-Gromov topology on compact subsets to a complete finite-volume hyperbolic three-manifold of sectional curvature $-1/4$. The present flow is three-dimensional, is a Ricci flow with surgery, and satisfies the required surgery-scale hypothesis $o(\sqrt{t})$. Choose $w_0$ from Perelman's theorem and then choose $T(w)$ from the same theorem. For every $w \in (0,w_0)$ and every nonsurgery sequence $t_k \geq T(w)$ with $t_k \to \infty$, the definition of $M_{\mathrm{thick}}(t_k,w)$ above is exactly Perelman's condition \begin{align*} \operatorname{Vol}_{g(t_k)}(B_{g(t_k)}(x_k,\rho(x_k,t_k))) \geq w\rho(x_k,t_k)^3. \end{align*} Hence Perelman's theorem applies to each thick basepoint sequence and gives the asserted subsequential hyperbolic limit after rescaling by $t_k^{-1}$. [guided] The point of using $\rho(x,t)$ is that Perelman's long-time theorem is stated at this specific curvature scale, not at an arbitrary scale chosen after the fact. For a point $x_k \in M_{\mathrm{thick}}(t_k,w)$, the definition says precisely that \begin{align*} \operatorname{Vol}_{g(t_k)}(B_{g(t_k)}(x_k,\rho(x_k,t_k))) \geq w\rho(x_k,t_k)^3. \end{align*} This is the noncollapsing alternative in Perelman's curvature-scale thick-thin theorem. We now verify the hypotheses of Perelman's Long-Time Thick-Thin Theorem for Ricci Flow with Surgery. The theorem requires a three-dimensional Ricci flow with surgery; this is part of the present theorem. It requires surgeries whose surgery scale is $o(\sqrt{t})$ as $t \to \infty$; this is the stated surgery-scale hypothesis, and it is what ensures that the surgeries disappear at the rescaled long-time scale. It also supplies constants $w_0>0$ and $T(w)>0$ so that the conclusion holds for every $w \in (0,w_0)$ and every nonsurgery time $t \geq T(w)$. Let $(t_k)_{k=1}^{\infty}$ be any sequence of nonsurgery times with $t_k \to \infty$, and let $x_k \in M_{\mathrm{thick}}(t_k,w)$ for each $k$. Since $t_k \to \infty$, after discarding finitely many indices we have $t_k \geq T(w)$. Perelman's theorem applies to the pointed rescaled manifolds $(M(t_k),t_k^{-1}g(t_k),x_k)$ and yields a subsequence converging smoothly in the pointed Cheeger-Gromov topology on compact subsets. The limit is a complete finite-volume hyperbolic three-manifold, and with the normalization $\tilde g_{t_k}=t_k^{-1}g(t_k)$ its sectional curvature is $-1/4$. This proves the exact sequential thick-part assertion: every pointed sequence based at $w$-thick points has a smoothly convergent subsequence to the required hyperbolic model. [/guided] [/step] [step:Use the rho-scale complement to obtain the locally collapsed thin part] By definition, a point $x \in M_{\mathrm{thin}}(t,w)$ is not $w$-thick at the curvature scale. Therefore \begin{align*} \operatorname{Vol}_{g(t)}(B_{g(t)}(x,\rho(x,t))) < w\rho(x,t)^3. \end{align*} The same constants $w_0$ and $T(w)$ from Perelman's Long-Time Thick-Thin Theorem for Ricci Flow with Surgery apply here, and for $t \geq T(w)$ its thin-part conclusion is stated exactly for the above $\rho(x,t)$-scale inequality. Combined with the three-dimensional collapse theory under lower sectional-curvature bounds, this gives local collapse on the thin part under a lower sectional-curvature bound. Therefore $M_{\mathrm{thin}}(t,w)$ is locally collapsed in the sense claimed. [/step] [step:Combine the two rho-scale alternatives into the required decomposition] For $w \in (0,w_0)$ and every nonsurgery time $t \geq T(w)$, the sets $M_{\mathrm{thick}}(t,w)$ and $M_{\mathrm{thin}}(t,w)$ are complementary subsets of $M(t)$, so they give a decomposition of the time-$t$ manifold into a $w$-thick part and a $w$-thin part. The preceding steps prove, respectively, the sequential hyperbolic finite-volume description of rescaled thick basepoint sequences and the local collapsed description of the thin part under a lower sectional-curvature bound. This is the asserted thick-thin decomposition for all sufficiently large nonsurgery times $t$ and every sufficiently small $w>0$. [/step]