[proofplan] We pass the Type I curvature estimate through the parabolic rescalings to obtain an ancient pointed smooth limit with uniformly controlled curvature on compact negative time intervals. Perelman's noncollapsing prevents collapse in the pointed Cheeger-Gromov limit. Finally, the Type I singular reduced-volume theory, based at the limiting spacetime point $(p,T)$ and passed to the blow-up limit, gives constant reduced volume on the nonflat limit; the equality case is precisely the gradient shrinking Ricci soliton equation. [/proofplan]

[step:Transfer the Type I curvature bound to the rescaled flows]Let $C_I \in (0,\infty)$ denote a Type I curvature constant for the original compact Ricci flow, so that \begin{align*} \sup_{x \in M} |\operatorname{Rm}_{g(t)}(x)|_{g(t)} \leq \frac{C_I}{T-t} \end{align*} for every $t \in [0,T)$. For each $j \in \mathbb{N}$, define the rescaled time interval \begin{align*} I_j := \{s \in \mathbb{R} : t_j + s/\lambda_j \in [0,T)\} = [-\lambda_j t_j,1). \end{align*} Define the map $g_j: I_j \to \Gamma(S^2T^*M)$ by \begin{align*} g_j(s)=\lambda_j g\left(t_j + \frac{s}{\lambda_j}\right) \end{align*} for $s\in I_j$. This map is again a Ricci flow, because parabolic scaling preserves the Ricci flow equation. Its curvature tensor satisfies the scaling identity \begin{align*} |\operatorname{Rm}_{g_j(s)}(x)|_{g_j(s)} = \lambda_j^{-1}|\operatorname{Rm}_{g(t_j+s/\lambda_j)}(x)|_{g(t_j+s/\lambda_j)}. \end{align*} Since $\lambda_j=(T-t_j)^{-1}$, we have \begin{align*} T-\left(t_j+\frac{s}{\lambda_j}\right)=T-t_j-\frac{s}{\lambda_j}=\frac{1-s}{\lambda_j}. \end{align*} Therefore, for every $s \in I_j$, \begin{align*} \sup_{x \in M}|\operatorname{Rm}_{g_j(s)}(x)|_{g_j(s)} \leq \lambda_j^{-1}\frac{C_I}{(1-s)/\lambda_j} =\frac{C_I}{1-s}. \end{align*} In particular, for each compact interval $[a,b] \subset (-\infty,0)$, the curvatures of $g_j(s)$ are uniformly bounded by $C_I/(1-b)$ on $M \times [a,b]$ for all sufficiently large $j$.[/step]

[guided]The first task is to check that the blow-up sequence has estimates that survive as $j \to \infty$. Let $C_I \in (0,\infty)$ be a Type I constant for the original flow, meaning \begin{align*} \sup_{x \in M} |\operatorname{Rm}_{g(t)}(x)|_{g(t)} \leq \frac{C_I}{T-t} \end{align*} for all $t \in [0,T)$. For each $j$, the rescaled flow is defined on the set of times for which the unscaled time remains in $[0,T)$: \begin{align*} I_j := \{s \in \mathbb{R} : t_j+s/\lambda_j \in [0,T)\}= [-\lambda_j t_j,1). \end{align*} We regard it as the map $g_j: I_j \to \Gamma(S^2T^*M)$ defined by \begin{align*} g_j(s)=\lambda_j g\left(t_j+\frac{s}{\lambda_j}\right) \end{align*} for $s\in I_j$. Parabolic scaling preserves the Ricci flow equation, so each $g_j$ is a Ricci flow. Curvature scales inversely with the metric scaling factor, hence \begin{align*} |\operatorname{Rm}_{g_j(s)}(x)|_{g_j(s)}=\lambda_j^{-1}|\operatorname{Rm}_{g(t_j+s/\lambda_j)}(x)|_{g(t_j+s/\lambda_j)}. \end{align*} Now the special choice $\lambda_j=(T-t_j)^{-1}$ is used. The remaining time to the singular time $T$ becomes \begin{align*} T-\left(t_j+\frac{s}{\lambda_j}\right)=T-t_j-\frac{s}{\lambda_j}=\frac{1-s}{\lambda_j}. \end{align*} Substituting this into the Type I estimate gives \begin{align*} \sup_{x \in M}|\operatorname{Rm}_{g_j(s)}(x)|_{g_j(s)} \leq \lambda_j^{-1}\frac{C_I}{(1-s)/\lambda_j} =\frac{C_I}{1-s}. \end{align*} Thus every compact interval $[a,b] \subset (-\infty,0)$ has a uniform curvature bound $C_I/(1-b)$ for all large $j$. This is exactly the scale-invariant curvature control needed to pass to an ancient smooth limit.[/guided]

[step:Use the assumed pointed convergence to obtain an ancient Ricci flow limit] Since $t_j \uparrow T$, we have $\lambda_j \to \infty$, and hence $-\lambda_j t_j \to -\infty$ after discarding finitely many indices. Therefore the intervals $I_j$ exhaust $(-\infty,1)$ from the left. By the assumed pointed Cheeger-Gromov subconvergence on compact time intervals $s \in (-\infty,0)$, after passing to the given subsequence there exist a pointed smooth manifold $(M_\infty,p_\infty)$, an ancient Ricci flow \begin{align*} g_\infty: (-\infty,0) &\to \Gamma(S^2T^*M_\infty), \end{align*} and pointed Cheeger-Gromov convergence \begin{align*} (M,g_j(s),p) \longrightarrow (M_\infty,g_\infty(s),p_\infty) \end{align*} smoothly on compact subsets of space-time. The curvature estimate from the previous step passes to the limit, giving \begin{align*} \sup_{x \in K}|\operatorname{Rm}_{g_\infty(s)}(x)|_{g_\infty(s)} \leq \frac{C_I}{1-s} \end{align*} for every compact set $K \subset M_\infty$ and every $s \in (-\infty,0)$. [/step]

[step:Apply noncollapsing to exclude collapsed ancient limits] Because the original flow is compact on the finite time interval $[0,T)$, Perelman's [No Local Collapsing Theorem](/theorems/6020) applies to $g(t)$: for every finite curvature scale below $T$, there is a constant $\kappa \in (0,\infty)$ such that metric balls whose curvature is bounded by the reciprocal square of the radius have volume at least $\kappa$ times the Euclidean model volume. This hypothesis is invariant under the parabolic rescaling used to define $g_j$. Passing to the smooth pointed Cheeger-Gromov limit preserves the lower volume ratio on compact balls, so the ancient limit $(M_\infty,g_\infty(s),p_\infty)$ is $\kappa$-noncollapsed on all scales on which its curvature is bounded. [/step]

[step:Pass Perelman's reduced-volume monotonicity to the blow-up limit]For each $j$, let $\ell_j: M \times (-\lambda_j t_j,1) \to \mathbb{R}$ denote Perelman's reduced distance for the rescaled flow $g_j(s)$ based at the spacetime point corresponding to $(p,T)$, namely $(p,1)$ in the $s$-coordinate. Define the backward-time parameter $\tau_j(s):=1-s$ for $s<1$. Let $\widetilde V_j: (-\lambda_j t_j,1) \to \mathbb{R}$ be the corresponding reduced-volume function, given by \begin{align*} \widetilde V_j(s)=\int_M (4\pi(1-s))^{-n/2}e^{-\ell_j(x,s)}\,d\operatorname{vol}_{g_j(s)}(x). \end{align*} Here $n=\dim M$ and $d\operatorname{vol}_{g_j(s)}$ is the Riemannian volume measure of $g_j(s)$. Perelman's [Reduced Volume Monotonicity Formula](/theorems/6018) applies because each $g_j$ is a smooth complete Ricci flow on compact time subintervals and $M$ is compact; it gives that $\widetilde V_j$ is nonincreasing as the backward time $\tau_j=1-s$ increases, and $0\leq \widetilde V_j\leq 1$. The reduced distance and reduced volume are invariant under the parabolic rescaling defining $g_j$, with base time $T$ in the original variables corresponding to base time $1$ in the rescaled variables. Thus, for fixed $s_1<s_2<0$, the quantity \begin{align*} \widetilde V_j(s_2)-\widetilde V_j(s_1) \end{align*} is the reduced-volume drop for the original flow between the backward times $(1-s_2)/\lambda_j$ and $(1-s_1)/\lambda_j$, both of which tend to $0$ as $j\to\infty$. Since the original reduced volume based at $(p,T)$ is monotone and bounded in $[0,1]$, it has a finite limit as the backward time tends to $0$. Hence \begin{align*} \lim_{j\to\infty}\left(\widetilde V_j(s_2)-\widetilde V_j(s_1)\right)=0. \end{align*} We use the reduced-distance compactness and reduced-volume convergence theorem for Type I pointed Ricci-flow blow-ups. Its hypotheses are satisfied here: the basepoints are fixed at $p$ and converge to $p_\infty$ under the pointed Cheeger-Gromov convergence, the preceding Type I estimate gives uniform curvature bounds on every compact negative time interval, Perelman's noncollapsing gives the required local volume lower bounds, and the Gaussian lower bounds for reduced distance give uniform tails for the weighted densities outside large pointed balls. Therefore $\ell_j$ subconverges locally locally uniformly to the reduced distance $\ell_\infty: M_\infty\times(-\infty,0)\to\mathbb{R}$ based at the limiting spacetime point $(p_\infty,1)$, and the weighted global integrals converge by local smooth convergence together with the uniform tail estimate. The reduced volume $\widetilde V_\infty: (-\infty,0)\to\mathbb{R}$ of $(M_\infty,g_\infty(s))$ based at $(p_\infty,1)$ therefore satisfies \begin{align*} \widetilde V_\infty(s_1)=\widetilde V_\infty(s_2) \end{align*} for every $s_1<s_2<0$.[/step]

[guided]We now identify the rigidity mechanism. The quantity that detects shrinking solitons is Perelman's reduced volume, but there is an important endpoint issue. The spacetime point $(p,T)$ is not a regular point of the original flow, and after rescaling it corresponds to $(p,1)$, while $g_j$ is defined only for $s<1$. Thus we cannot simply apply the ordinary reduced-distance construction with a regular basepoint. We instead use the [Type I Singular Reduced Distance and Reduced Volume Convergence Theorem](/theorems/6021), which defines the singular reduced distance by taking regular base times tending to the Type I endpoint, proves monotonicity for the resulting singular reduced volume, and proves convergence of these quantities under pointed Type I blow-up. For each rescaled flow $g_j$, let $\ell_j: M\times(-\lambda_jt_j,1)\to\mathbb{R}$ denote this singular reduced distance with boundary base point $(p,1)$ in the rescaled time coordinate. This is the same spacetime endpoint as $(p,T)$ for the original flow, because $s=\lambda_j(t-t_j)$ sends $t=T$ to $s=1$. The backward-time parameter from the boundary base time $1$ to a time $s<1$ is not $-s$; it is \begin{align*} \tau_j(s)=1-s. \end{align*} Thus the corresponding singular reduced volume is the function $\widetilde V_j: (-\lambda_jt_j,1)\to\mathbb{R}$ defined by \begin{align*} \widetilde V_j(s)=\int_M (4\pi(1-s))^{-n/2}e^{-\ell_j(x,s)}\,d\operatorname{vol}_{g_j(s)}(x), \end{align*} where $n=\dim M$ and $d\operatorname{vol}_{g_j(s)}$ is the Riemannian volume measure of the metric $g_j(s)$. This is the point where the base-time bookkeeping matters: the factor $(4\pi(1-s))^{-n/2}$ is forced by the boundary base point $(p,1)$. The hypotheses of the [Type I Singular Reduced Distance and Reduced Volume Convergence Theorem](/theorems/6021) are satisfied. Each $g_j$ is smooth and complete on compact time subintervals, because $M$ is compact and parabolic rescaling preserves smoothness and completeness. The Type I curvature estimate gives the required endpoint curvature control as $s\uparrow 1$. The theorem therefore gives monotonicity in the backward-time variable $\tau_j=1-s$ and the bound $0\leq \widetilde V_j\leq 1$. Why should the limit have constant reduced volume? Fix $s_1<s_2<0$. Under the inverse rescaling, these times correspond to \begin{align*} t_{j,k}:=t_j+\frac{s_k}{\lambda_j}, \qquad k\in\{1,2\}. \end{align*} The corresponding backward times from $T$ are \begin{align*} T-t_{j,k}=\frac{1-s_k}{\lambda_j}, \qquad k\in\{1,2\}, \end{align*} and both tend to $0$ as $j\to\infty$. Since the original singular reduced volume based at $(p,T)$ is monotone and bounded in $[0,1]$, it has a finite limit as the backward time tends to $0$. Parabolic scaling leaves singular reduced distance and singular reduced volume unchanged with this matching boundary base time, so the drop satisfies \begin{align*} \lim_{j\to\infty}\left(\widetilde V_j(s_2)-\widetilde V_j(s_1)\right)=0. \end{align*} It remains to justify passage of the global integral to the pointed limit. Local smooth Cheeger-Gromov convergence alone only gives convergence on fixed compact subsets, so we invoke the compactness and convergence part of the [Type I Singular Reduced Distance and Reduced Volume Convergence Theorem](/theorems/6021). Its assumptions are verified as follows: the basepoints are the fixed points $p$ and converge to $p_\infty$ in the pointed limit; the Type I estimate gives uniform curvature bounds on compact negative time intervals; Perelman's noncollapsing gives the local volume lower bounds; and the Gaussian lower estimates for singular reduced distance give uniform integrable tails for $(4\pi(1-s))^{-n/2}e^{-\ell_j}$ outside large pointed balls. The theorem yields local convergence of $\ell_j$ to the limiting singular reduced distance $\ell_\infty: M_\infty\times(-\infty,0)\to\mathbb{R}$ based at $(p_\infty,1)$, and the tail estimate upgrades local integral convergence to convergence of the full reduced volumes. Passing to the limit in the zero-drop identity gives \begin{align*} \widetilde V_\infty(s_1)=\widetilde V_\infty(s_2) \end{align*} for every $s_1<s_2<0$. Thus the singular reduced volume of the limit is constant on $(-\infty,0)$.[/guided]

[step:Invoke the equality case to obtain the shrinking soliton equation] By the equality case in Perelman's reduced-volume monotonicity, equivalently the [Reduced Volume Rigidity Theorem](/theorems/6019), a complete ancient Ricci flow with constant reduced volume on a time interval is a gradient shrinking Ricci soliton on that interval. Applying this result to $(M_\infty,g_\infty(s))$ yields a smooth potential function $f_\infty: M_\infty \times (-\infty,0) \to \mathbb{R}$ such that \begin{align*} \operatorname{Ric}_{g_\infty(s)} + \operatorname{Hess}_{g_\infty(s)} f_\infty(\cdot,s) = \frac{1}{2(1-s)}g_\infty(s) \end{align*} for every $s \in (-\infty,0)$, where $\operatorname{Hess}_{g_\infty(s)}$ denotes the Hessian with respect to the metric $g_\infty(s)$. Hence the pointed Cheeger-Gromov limit is an ancient gradient shrinking Ricci soliton. Under the theorem's additional assumption that the selected limit is nonflat, this proves that every nonflat subsequential blow-up limit based at $p$ is an ancient gradient shrinking Ricci soliton. [/step]