[proofplan]
We first normalize the type-I parabolic rescalings so that the singular time becomes $s=1$ and verify that both the curvature upper bound and the type-I lower bound pass to the rescaled sequence. The central analytic input is the precise Enders-Mueller-Topping blow-up theorem, in Naber's reduced-volume formulation: for a closed type-I Ricci flow, any pointed smooth limit obtained from a type-I blow-up sequence has terminal reduced volume constant in backward time. This theorem packages the singular reduced-distance construction, convergence of reduced-volume densities for moving base points, and the equality case of Perelman's monotonicity. Applying its rigidity conclusion gives the gradient shrinking Ricci soliton equation, and the rescaled curvature lower bound gives nonflatness.
[/proofplan]
[step:Normalize the parabolic rescalings and preserve the type-I curvature bound]
For each $j\in\mathbb{N}$ define the scale
\begin{align*}
\lambda_j := \frac{1}{T-t_j}
\end{align*}
and the time interval
\begin{align*}
I_j := [-\lambda_j t_j,1).
\end{align*}
Define the rescaled flow $g_j:I_j\to \Gamma(\operatorname{Sym}^2T^*M)$ by
\begin{align*}
g_j(s):=\lambda_j\,g(t_j+\lambda_j^{-1}s).
\end{align*}
Since $t_j\uparrow T$, we have $\lambda_j\to\infty$ and hence $-\lambda_j t_j\to-\infty$. Thus every compact interval $K\subset(-\infty,1)$ is contained in $I_j$ for all sufficiently large $j$.
Let $\operatorname{Rm}_{g_j(s)}$ denote the Riemann curvature tensor of $g_j(s)$. Curvature scales by
\begin{align*}
|\operatorname{Rm}|_{g_j(s)}(x)
=
\lambda_j^{-1}
|\operatorname{Rm}|_{g(t_j+\lambda_j^{-1}s)}(x).
\end{align*}
Using the type-I bound on the original flow and the identity
\begin{align*}
T-\left(t_j+\lambda_j^{-1}s\right)
=
(T-t_j)(1-s)
=
\lambda_j^{-1}(1-s),
\end{align*}
we obtain, for every $x\in M$ and $s\in I_j$,
\begin{align*}
|\operatorname{Rm}|_{g_j(s)}(x)
\leq
\lambda_j^{-1}\frac{C_0}{T-(t_j+\lambda_j^{-1}s)}
=
\frac{C_0}{1-s}.
\end{align*}
In particular, on each compact interval $K\subset(-\infty,1)$ the curvatures of $g_j(s)$ are uniformly bounded by
\begin{align*}
C_K:=\frac{C_0}{\inf_{s\in K}(1-s)}.
\end{align*}
At the base points,
\begin{align*}
|\operatorname{Rm}|_{g_j(0)}(p_j)
=
\lambda_j^{-1}|\operatorname{Rm}|_{g(t_j)}(p_j)
\geq
\lambda_j^{-1}\frac{c_0}{T-t_j}
=
c_0.
\end{align*}
Therefore any pointed smooth limit of the flows $(M,g_j(s),p_j)$ is nonflat at time $s=0$.
[guided]
The parabolic scaling is chosen so that the remaining time to the singularity becomes exactly $1$. For each $j$, the original time $t=t_j$ becomes rescaled time $s=0$, while the singular time $T$ becomes rescaled time $s=1$, since
\begin{align*}
T=t_j+\lambda_j^{-1}\cdot 1.
\end{align*}
Thus the natural backward time in the rescaled flow is $\tau:=1-s$.
We define
\begin{align*}
\lambda_j := \frac{1}{T-t_j},
\qquad
g_j(s):=\lambda_j\,g(t_j+\lambda_j^{-1}s),
\qquad
s\in[-\lambda_j t_j,1).
\end{align*}
The left endpoint tends to $-\infty$ because $t_j\to T>0$ and $\lambda_j\to\infty$. Hence the rescaled flows exist on longer and longer backward time intervals, and a limiting flow can be defined on all $s\in(-\infty,1)$.
Now we check how the type-I curvature estimate behaves under the scaling. If a Riemannian metric is multiplied by $\lambda_j$, then the norm of curvature is multiplied by $\lambda_j^{-1}$. Therefore
\begin{align*}
|\operatorname{Rm}|_{g_j(s)}(x)
=
\lambda_j^{-1}
|\operatorname{Rm}|_{g(t_j+\lambda_j^{-1}s)}(x).
\end{align*}
The original type-I hypothesis gives
\begin{align*}
|\operatorname{Rm}|_{g(t_j+\lambda_j^{-1}s)}(x)
\leq
\frac{C_0}{T-(t_j+\lambda_j^{-1}s)}.
\end{align*}
The denominator transforms exactly as
\begin{align*}
T-(t_j+\lambda_j^{-1}s)
=
(T-t_j)-\lambda_j^{-1}s
=
\lambda_j^{-1}(1-s).
\end{align*}
Substituting this into the curvature estimate yields
\begin{align*}
|\operatorname{Rm}|_{g_j(s)}(x)
\leq
\lambda_j^{-1}\frac{C_0}{\lambda_j^{-1}(1-s)}
=
\frac{C_0}{1-s}.
\end{align*}
This is the scale-invariant form of the type-I bound. On any compact time interval strictly below $s=1$, the denominator $1-s$ is bounded below by a positive number, so the curvatures are uniformly bounded.
Finally, the definition of a type-I blow-up sequence prevents the limit from becoming flat. At the base point and rescaled time $s=0$,
\begin{align*}
|\operatorname{Rm}|_{g_j(0)}(p_j)
=
\lambda_j^{-1}|\operatorname{Rm}|_{g(t_j)}(p_j)
\geq
\lambda_j^{-1}\frac{c_0}{T-t_j}
=
c_0.
\end{align*}
Smooth pointed convergence preserves curvature at the base point, so the limiting curvature at $(p_\infty,0)$ has norm at least $c_0$. Hence the pointed limit is nonflat.
[/guided]
[/step]
[step:Record the assumed pointed smooth blow-up limit]
By hypothesis, after passing to the subsequence defining the blow-up limit, there exist a smooth manifold $M_\infty$, a point $p_\infty\in M_\infty$, and a complete Ricci flow
\begin{align*}
g_\infty:(-\infty,1)\to\Gamma(\operatorname{Sym}^2T^*M_\infty)
\end{align*}
such that
\begin{align*}
(M,g_j(s),p_j)\to(M_\infty,g_\infty(s),p_\infty)
\end{align*}
smoothly in the pointed Cheeger-Gromov sense on compact subsets of space-time. The curvature estimate from the previous step passes to the limit and gives, for every $y\in M_\infty$ and $s\in(-\infty,1)$,
\begin{align*}
|\operatorname{Rm}|_{g_\infty(s)}(y)\leq \frac{C_0}{1-s}.
\end{align*}
The base-point curvature lower bound also passes to the pointed smooth limit, so
\begin{align*}
|\operatorname{Rm}|_{g_\infty(0)}(p_\infty)\geq c_0.
\end{align*}
Thus the limiting flow is complete, has bounded curvature on compact time intervals, and is nonflat.
[/step]
[step:Invoke the type-I blow-up theorem in reduced-volume form]
Let $n:=\dim M=\dim M_\infty$ denote the common manifold dimension, where equality follows from pointed smooth Cheeger-Gromov convergence. For each $j\in\mathbb{N}$, let $\tau:=1-s$ denote the backward time for the rescaled flow.
We invoke the Enders-Mueller-Topping type-I blow-up theorem, equivalently Naber's reduced-volume rigidity formulation for type-I singularities, in the following form. If a closed Ricci flow satisfies a type-I curvature bound and $(p_j,t_j)$ is a type-I blow-up sequence, then every pointed smooth parabolic blow-up limit based at $(p_j,t_j)$ admits a terminal reduced distance
\begin{align*}
\ell_\infty:M_\infty\times(0,\infty)\to\mathbb{R}
\end{align*}
based at the limiting spacetime point $(p_\infty,1)$. For this terminal reduced distance, the reduced-volume function defined for $\tau\in(0,\infty)$ by
\begin{align*}
\tilde{V}_\infty(\tau):=\int_{M_\infty}(4\pi\tau)^{-n/2}e^{-\ell_\infty(y,\tau)}\,d\operatorname{vol}_{g_\infty(1-\tau)}(y)
\end{align*}
is constant in $\tau$, and the equality case in Perelman's reduced-volume monotonicity yields a smooth function
\begin{align*}
f:M_\infty\times(-\infty,1)\to\mathbb{R}
\end{align*}
solving the gradient shrinking soliton equation.
The theorem applies here because $M$ is closed, the original curvature estimate is exactly the type-I bound, the sequence $(p_j,t_j)$ satisfies the type-I lower bound, and the previous step records pointed smooth convergence of the rescaled flows with locally bounded curvature on compact time intervals.
[/step]
[step:Use constancy of the limiting reduced volume]
Let $\ell_\infty:M_\infty\times(0,\infty)\to\mathbb{R}$ and $\tilde{V}_\infty:(0,\infty)\to\mathbb{R}$ be the terminal reduced distance and reduced volume supplied by the type-I blow-up theorem in the preceding step. That theorem states directly that $\tilde{V}_\infty$ is constant in backward time. Therefore, for every pair $0<\tau_1<\tau_2<\infty$,
\begin{align*}
\tilde{V}_\infty(\tau_1)=\tilde{V}_\infty(\tau_2).
\end{align*}
[guided]
The point of invoking the type-I blow-up theorem is that it supplies the analytic limiting argument as a theorem rather than requiring us to reprove it inside this proof. The theorem does more than ordinary monotonicity: ordinary reduced-volume monotonicity controls a fixed reduced volume, while the type-I blow-up theorem handles the moving base points $(p_j,t_j)$, the passage to singular terminal reduced distances, the convergence of reduced-volume densities on the pointed limit, and the equality case. Thus we are not using the false inference that two quantities must have the same limit merely because their backward times both tend to $0$.
With the notation from the preceding step, the theorem gives a terminal reduced distance
\begin{align*}
\ell_\infty:M_\infty\times(0,\infty)\to\mathbb{R}
\end{align*}
and the associated reduced volume
\begin{align*}
\tilde{V}_\infty(\tau)
=
\int_{M_\infty}
(4\pi\tau)^{-n/2}e^{-\ell_\infty(y,\tau)}
\,d\operatorname{vol}_{g_\infty(1-\tau)}(y).
\end{align*}
Its conclusion is exactly the constancy statement
\begin{align*}
\tilde{V}_\infty(\tau_1)=\tilde{V}_\infty(\tau_2)
\end{align*}
for all $0<\tau_1<\tau_2<\infty$.
[/guided]
[/step]
[step:Apply the rigidity conclusion of the type-I blow-up theorem]
The rigidity conclusion in the Enders-Mueller-Topping theorem states that the pointed smooth blow-up limit of a closed type-I Ricci flow at a type-I blow-up sequence is a normalized gradient shrinking Ricci soliton. In reduced-volume language, this is the equality case of Perelman's reduced-volume monotonicity for the terminal reduced distance supplied by the theorem.
The limiting flow $g_\infty(s)$ is complete and has bounded curvature on compact time intervals by the previous steps. The preceding step gives constancy of the terminal reduced volume on all $\tau\in(0,\infty)$, and the type-I blow-up theorem supplies the analytic hypotheses needed for the equality case, including the terminal reduced distance, smoothness of the limiting potential, and the noncompact exhaustion justified by Gaussian decay of the reduced-volume density. Hence there exists a smooth potential function
\begin{align*}
f:M_\infty\times(-\infty,1)\to\mathbb{R}
\end{align*}
such that, for every $s\in(-\infty,1)$,
\begin{align*}
\operatorname{Ric}_{g_\infty(s)}
+
\nabla^{g_\infty(s)}d(f(\cdot,s))
=
\frac{1}{2(1-s)}g_\infty(s).
\end{align*}
This is the gradient shrinking Ricci soliton equation in canonical backward-time normalization, written using the covariant derivative of the one-form $d(f(\cdot,s))$ instead of forbidden second-gradient notation. Therefore every nonflat pointed smooth blow-up limit obtained from the type-I sequence is a gradient shrinking Ricci soliton.
[guided]
The final step is the rigidity step: constancy of the terminal reduced volume is not merely an estimate, but the equality case of Perelman's reduced-volume monotonicity as supplied by the type-I blow-up theorem. We have already verified the hypotheses needed to use that conclusion: the limiting flow
\begin{align*}
g_\infty:(-\infty,1)\to\Gamma(\operatorname{Sym}^2T^*M_\infty)
\end{align*}
is complete, has bounded curvature on compact time intervals, and arises as a pointed smooth blow-up limit of a closed type-I Ricci flow at a type-I blow-up sequence.
The theorem therefore provides a smooth potential function
\begin{align*}
f:M_\infty\times(-\infty,1)\to\mathbb{R}
\end{align*}
such that, for every $s\in(-\infty,1)$,
\begin{align*}
\operatorname{Ric}_{g_\infty(s)}
+
\nabla^{g_\infty(s)}d(f(\cdot,s))
=
\frac{1}{2(1-s)}g_\infty(s).
\end{align*}
This equation is precisely the gradient shrinking Ricci soliton equation in canonical backward-time normalization, with singular time normalized to $s=1$. The tensor $\nabla^{g_\infty(s)}d(f(\cdot,s))$ is the Hessian of the potential written as the covariant derivative of the one-form $d(f(\cdot,s))$. Since the base-point curvature lower bound passed to the limit, the soliton is nonflat. Hence the pointed smooth blow-up limit is a nonflat gradient shrinking Ricci soliton.
[/guided]
[/step]
