[proofplan] The proof is the standard contradiction and blow-up argument in Perelman's canonical-neighbourhood theory. [Finite-time no-local-collapsing](/theorems/6024) supplies a uniform $\kappa>0$, Hamilton-Ivey pinching forces the high-curvature blow-up limits to have nonnegative curvature operator, and [Hamilton compactness](/theorems/6004) extracts an ancient pointed limit. The classification and [local structure theorem for three-dimensional ancient $\kappa$-solutions](/theorems/6010) then gives exactly the canonical-neighbourhood alternatives, while the scale-invariant derivative estimates follow from the same ancient-solution model estimates after rescaling back. [/proofplan]

[step:Fix the noncollapsing and pinching constants supplied by the initial data]Let $\kappa=\kappa(g(0),T)>0$ be the constant given by Perelman's finite-time no-local-collapsing theorem for the closed Ricci flow $(M^3,g(t))$, $t\in[0,T)$. Its hypotheses apply because $M$ is closed, the flow is smooth on the finite time interval $[0,T)$, the initial curvature is bounded by $|\operatorname{Rm}|_{g(0)}\leq 1$, and the initial unit-ball volume lower bound is $v_0>0$. Thus there is a scale $\rho_0=\rho_0(g(0),T)>0$ such that every parabolic ball of spatial radius at most $\rho_0$ on which $|\operatorname{Rm}|$ is bounded by the inverse square of that radius is $\kappa$-noncollapsed at that scale. This scale restriction is harmless in the blow-up argument, because every fixed radius in a normalized limit corresponds in the original flow to radius tending to $0$. Let $\nu=\nu(g(0),T)$ denote the Hamilton-Ivey pinching function supplied by the [Hamilton-Ivey pinching estimate](/theorems/6022). Since $|\operatorname{Rm}|_{g(0)}\leq 1$ and the flow is three-dimensional, the estimate applies on $[0,T)$ and implies that after rescaling around points with scalar curvature tending to infinity, every smooth pointed limit has nonnegative curvature operator.[/step]

[guided]The constants needed later come from two global inputs. First, Perelman's finite-time no-local-collapsing theorem applies to this closed smooth Ricci flow on the finite interval $[0,T)$ because the initial metric is normalized by $|\operatorname{Rm}|_{g(0)}\leq 1$ and every unit ball has volume at least $v_0>0$. We therefore obtain a number \begin{align*} \kappa=\kappa(g(0),T)>0. \end{align*} This means that any parabolic ball with curvature bounded by the reciprocal square of its spatial radius has volume at least $\kappa$ times the Euclidean model volume at that scale. Second, Hamilton-Ivey pinching applies because the solution is a three-dimensional Ricci flow with normalized initial curvature. It gives a pinching function $\nu=\nu(g(0),T)$ controlling the negative part of the curvature operator in terms of scalar curvature. The point of this estimate is that when we blow up around points with scalar curvature $Q_i\to\infty$, the negative part is multiplied by $Q_i^{-1}$ and tends to zero. Hence every smooth pointed blow-up limit has nonnegative curvature operator.[/guided]

[step:Assume a bad sequence in the fixed flow and rescale at its curvature scale]Suppose the asserted threshold does not exist for the fixed Ricci flow $(M^3,g(t))$, $t\in[0,T)$. Define the scalar-curvature function \begin{align*} R_g:M\times[0,T)&\to\mathbb{R} \end{align*} by letting $R_g(x,t)$ be the scalar curvature of the metric $g(t)$ at the point $x\in M$. Then for each integer $i\geq 1$ there is a point $(x_i,t_i)\in M\times[0,T)$ such that \begin{align*} Q_i:=R_g(x_i,t_i)\geq i^2 \end{align*} and $(x_i,t_i)$ has no $(\varepsilon,A,k)$-canonical neighbourhood at scale $Q_i^{-1/2}$. Thus $Q_i\to\infty$ in the same flow; the constants fixed above, including $\kappa$ and the Hamilton-Ivey pinching function, are the constants of this one initial metric. For each $i$, define the rescaled time interval $I_i\subset\mathbb{R}$ by \begin{align*} I_i:=\{s\in\mathbb{R}:t_i+Q_i^{-1}s\in[0,T)\}=[-Q_i t_i,Q_i(T-t_i)). \end{align*} Let $\Gamma(S^2T^*M)$ denote the smooth symmetric covariant two-tensors on $M$. Define the parabolically rescaled Ricci flow $\widetilde g_i:I_i\to\Gamma(S^2T^*M)$ by \begin{align*} \widetilde g_i(s):=Q_i\,g(t_i+Q_i^{-1}s). \end{align*} Then \begin{align*} R_{\widetilde g_i}(x_i,0)=1. \end{align*} The no-local-collapsing constant remains $\kappa$ under this parabolic rescaling, and Hamilton-Ivey pinching implies that the negative part of the curvature operator of $\widetilde g_i(0)$ tends to zero on bounded-curvature pointed regions. Here a parabolic ball $P_{g}(x,t,\rho,-\rho^2)$ means the spacetime set of points $(y,s)$ with $s\in[t-\rho^2,t]$ and $d_{g(t)}(x,y)<\rho$, using the standard comparison of nearby time-slice distances under a curvature bound. A strong $\varepsilon$-neck is a pointed parabolic neighbourhood which, after scaling the base scalar curvature to $1$, is $C^k$-$\varepsilon$-close on the prescribed time interval to the corresponding region in the shrinking round cylinder $S^2\times\mathbb{R}$. An $\varepsilon$-cap is a pointed high-curvature region whose compact core is attached to one such neck. An ancient $\kappa$-solution is a complete ancient three-dimensional Ricci flow with bounded nonnegative curvature operator on compact time intervals, positive scalar curvature, and $\kappa$-noncollapsing at all scales. Pointed $C^k$ parabolic convergence means $C^k$ convergence, after pulling back by pointed spacetime diffeomorphisms, of the metrics and their time derivatives on each compact pointed parabolic region.[/step]

[guided]We argue by contradiction inside the single Ricci flow appearing in the theorem. This matters because the theorem allows $r$ and $C$ to depend on the particular initial metric $g(0)$ and on the corresponding lower volume information $v_0$; therefore a failure of the theorem does not produce unrelated flows. It produces points in this fixed spacetime with scalar curvature tending to infinity. Thus, for each integer $i\geq 1$, choose $(x_i,t_i)\in M\times[0,T)$ such that \begin{align*} Q_i:=R_g(x_i,t_i)\geq i^2 \end{align*} and such that $(x_i,t_i)$ has no $(\varepsilon,A,k)$-canonical neighbourhood at scale $Q_i^{-1/2}$. Hence $Q_i\to\infty$ while the noncollapsing and pinching constants remain the constants already fixed for this one flow. The natural way to inspect the geometry at $(x_i,t_i)$ is to make its scalar curvature equal to $1$. Define \begin{align*} I_i:=\{s\in\mathbb{R}:t_i+Q_i^{-1}s\in[0,T)\}=[-Q_i t_i,Q_i(T-t_i)). \end{align*} Let $\Gamma(S^2T^*M)$ denote the smooth symmetric covariant two-tensors on $M$. Now define the rescaled flow as the map $\widetilde g_i:I_i\to\Gamma(S^2T^*M)$ given by \begin{align*} \widetilde g_i(s):=Q_i\,g(t_i+Q_i^{-1}s). \end{align*} Under this scaling scalar curvature is multiplied by $Q_i^{-1}$, so \begin{align*} R_{\widetilde g_i}(x_i,0)=Q_i^{-1}R_g(x_i,t_i)=1. \end{align*} The definition of $\kappa$-noncollapsing is invariant under simultaneous metric and parabolic time rescaling, so the same $\kappa$ applies to $\widetilde g_i$. Hamilton-Ivey pinching is also compatible with this blow-up limit: when curvature is scaled by $Q_i^{-1}$ and $Q_i\to\infty$, the negative part allowed by the pinching estimate disappears on bounded-curvature regions.[/guided]

[step:Use point selection and compactness to obtain an ancient noncollapsed limit]First $Q_i t_i\to\infty$. Indeed, the normalized curvature bound $|\operatorname{Rm}|_{g(0)}\leq1$ and closedness give a short-time curvature estimate: there are constants $\tau=\tau(g(0))>0$ and $B=B(g(0))<\infty$ such that $|\operatorname{Rm}|_{g(t)}\leq B$ on $M\times[0,\tau]$. Since $Q_i\to\infty$, the times $t_i$ cannot remain in an interval of size $O(Q_i^{-1})$ near $0$; otherwise $Q_i=R_g(x_i,t_i)\leq 3B$ for all large $i$. Hence the left endpoints $-Q_i t_i$ of the rescaled intervals tend to $-\infty$. Apply Perelman's point-selection lemma to the bad set of spacetime points which fail the $(\varepsilon,A,k)$-canonical-neighbourhood conclusion. In the form used here, the lemma says that from any sequence of bad points with scalar curvature tending to infinity one may choose bad points, still denoted $(x_i,t_i)$, with $Q_i=R_g(x_i,t_i)\to\infty$, and numbers $L_i\to\infty$, such that \begin{align*} |\operatorname{Rm}|_{\widetilde g_i}\leq B \end{align*} on $P_{\widetilde g_i}(x_i,0,L_i,-L_i)$ for one constant $B<\infty$ independent of $i$. The hypotheses apply because the bad set is closed under taking the selected replacement point in the backward high-curvature search region, the scalar curvature at the original bad points tends to infinity, and the canonical-neighbourhood property is scale invariant. The selection is made within the bad set, so no change of $\varepsilon$ is made and the contradiction property is preserved exactly. Since $L_i\to\infty$, this gives a uniform curvature bound on every compact pointed parabolic region and, after passage to the limit, a global bound on each time-slice of the limiting ancient flow. Hamilton compactness for pointed Ricci flows applies to \begin{align*} (M,\widetilde g_i(s),x_i). \end{align*} Its hypotheses are satisfied: the point-selection lemma gives uniform curvature bounds on every compact pointed parabolic neighbourhood, Shi estimates give the corresponding higher derivative bounds, the Ricci-flow equation is satisfied by each $\widetilde g_i$, and the injectivity-radius lower bound at the base points follows from $\kappa$-noncollapsing together with bounded curvature. Passing to a subsequence, the pointed flows converge in $C^\infty_{\mathrm{loc}}$ to a complete ancient pointed Ricci flow \begin{align*} (M_\infty,g_\infty(s),x_\infty),\qquad s\in(-\infty,0], \end{align*} with \begin{align*} R_{g_\infty}(x_\infty,0)=1. \end{align*} For any fixed scale in the limit, the corresponding scale in the original unrescaled flow is multiplied by $Q_i^{-1/2}$ and is therefore eventually below the finite-time noncollapsing scale $\rho_0$. Hence the limit is $\kappa$-noncollapsed at all scales. Hamilton-Ivey pinching gives nonnegative curvature operator in the limit because the negative curvature-operator part vanishes after multiplication by $Q_i^{-1}$. The uniform point-selection bound on balls whose radii tend to infinity also implies that, for each $s\leq0$, the curvature of $g_\infty(s)$ is bounded on all of $M_\infty$. Since $R_{g_\infty}(x_\infty,0)=1$ and the curvature operator is nonnegative, the limit has positive scalar curvature at the base point; the strong maximum principle for Ricci flow then gives positive scalar curvature everywhere on the nonflat ancient solution.[/step]

[guided]There are two compactness issues to settle before taking a limit. The first is the time interval. The rescaled flow is defined for \begin{align*} s\in[-Q_i t_i,Q_i(T-t_i)). \end{align*} To obtain an ancient limit, the left endpoint must tend to $-\infty$. The normalized initial curvature bound and smooth short-time existence on the closed manifold give constants $\tau=\tau(g(0))>0$ and $B=B(g(0))<\infty$ with \begin{align*} |\operatorname{Rm}|_{g(t)}\leq B\quad\text{on }M\times[0,\tau]. \end{align*} If $Q_i t_i$ were bounded along a subsequence, then there would be a constant $\Lambda<\infty$ such that $t_i\leq \Lambda Q_i^{-1}$ along that subsequence, and hence $t_i\to0$. For large $i$ we would have $t_i\in[0,\tau]$. Since in dimension three the scalar curvature satisfies $|R_g|\leq 3|\operatorname{Rm}|_{g(t)}$, this would imply $Q_i=R_g(x_i,t_i)\leq 3B$, contradicting $Q_i\to\infty$. Therefore \begin{align*} Q_i t_i\to\infty. \end{align*} The second issue is curvature control on regions large enough to become the whole limiting solution. We use Perelman's point-selection lemma on the set of bad points, meaning the points that fail the desired $(\varepsilon,A,k)$-canonical-neighbourhood conclusion. The selected points remain bad, so the contradiction hypothesis is preserved exactly; there is no reduction of $\varepsilon$. The output is that, after rescaling by $Q_i=R_g(x_i,t_i)$, there are radii $L_i\to\infty$ and a constant $B<\infty$ such that \begin{align*} |\operatorname{Rm}|_{\widetilde g_i}\leq B \end{align*} on $P_{\widetilde g_i}(x_i,0,L_i,-L_i)$. Because the controlled radii tend to infinity, every point of every fixed time-slice of the limiting flow is reached inside these controlled regions for all sufficiently large $i$. Now Hamilton compactness applies. The curvature bounds give derivative bounds by the Ricci-flow evolution equations and Shi estimates; $\kappa$-noncollapsing plus bounded curvature gives a uniform injectivity-radius lower bound at the base points; and each $\widetilde g_i$ solves the Ricci-flow equation. Hence a subsequence of \begin{align*} (M,\widetilde g_i(s),x_i) \end{align*} converges smoothly on compact pointed parabolic subsets to a complete pointed Ricci flow \begin{align*} (M_\infty,g_\infty(s),x_\infty),\qquad s\in(-\infty,0]. \end{align*} The normalization passes to the limit: \begin{align*} R_{g_\infty}(x_\infty,0)=1. \end{align*} The noncollapsing condition passes to the limit by smooth convergence of balls with bounded curvature, and Hamilton-Ivey pinching passes to the limit because the negative curvature operator part vanishes under the blow-up. The strengthened point-selection bound gives bounded curvature on each complete time-slice of the limit, not merely on compact pointed balls. Since $R_{g_\infty}(x_\infty,0)=1$ and the curvature operator is nonnegative, the limit is nonflat and has positive scalar curvature at the base point; by the strong maximum principle for Ricci flow, its scalar curvature is positive everywhere. Thus the limit is a genuine three-dimensional ancient $\kappa$-solution.[/guided]

[step:Apply the ancient $\kappa$-solution structure theorem to contradict badness]The limit $(M_\infty,g_\infty(s))$, $s\in(-\infty,0]$, is complete and ancient. The strengthened point-selection bound gives bounded curvature on each time-slice, Hamilton-Ivey gives nonnegative curvature operator, $R_{g_\infty}(x_\infty,0)=1$ makes the solution nonflat, the strong maximum principle gives positive scalar curvature everywhere, and the finite-time no-local-collapsing argument gives $\kappa$-noncollapsing at all scales. Hence it is a three-dimensional ancient $\kappa$-solution. We use the three-dimensional ancient $\kappa$-solution canonical-neighbourhood theorem in the following form: for every $\varepsilon>0$, $A>0$, and integer $k\geq10$, every point of every three-dimensional ancient $\kappa$-solution with positive scalar curvature, after scaling its scalar curvature to $1$, lies in one of the standard $(\varepsilon,A,k)$ model neighbourhoods: the centre of a strong $\varepsilon$-neck, an $\varepsilon$-cap with cylindrical end, a compact positively curved component, or a pointed parabolic region $C^k$-close on the prescribed scale to an ancient $\kappa$-solution model. Its hypotheses are exactly the properties verified for the limit: completeness, ancientness, bounded curvature on each time-slice and hence on compact time intervals, nonnegative curvature operator, nonflatness, positive scalar curvature everywhere, and $\kappa$-noncollapsing at all scales. Applying this theorem at $(x_\infty,0)$ gives one of the listed $(\varepsilon,A,k)$ alternatives at scale $R_{g_\infty}(x_\infty,0)^{-1/2}=1$. Because $(M,\widetilde g_i(s),x_i)$ converges to $(M_\infty,g_\infty(s),x_\infty)$ smoothly on compact pointed parabolic sets, and because each alternative in the definition of an $(\varepsilon,A,k)$-canonical neighbourhood is open under sufficiently close pointed $C^k$ parabolic convergence, the corresponding point $(x_i,0)$ in the rescaled flow has an $(\varepsilon,A,k)$-canonical neighbourhood for all large $i$. Rescaling back, $(x_i,t_i)$ has an $(\varepsilon,A,k)$-canonical neighbourhood at scale $Q_i^{-1/2}$, contradicting the choice of the bad sequence.[/step]

[guided]The limit has exactly the ingredients required in the definition of an ancient $\kappa$-solution. It is ancient because its time interval is $(-\infty,0]$; it is complete by Hamilton compactness; it has bounded curvature on each time-slice because the point-selection bound holds on parabolic balls whose radii tend to infinity; it has nonnegative curvature operator by Hamilton-Ivey pinching; it is nonflat because \begin{align*} R_{g_\infty}(x_\infty,0)=1; \end{align*} and it is $\kappa$-noncollapsed at all scales because noncollapsing is stable under smooth pointed convergence. The scalar curvature is positive at the base point, and the strong maximum principle for Ricci flow promotes this to positive scalar curvature everywhere on the nonflat ancient solution with nonnegative curvature operator. Therefore $(M_\infty,g_\infty(s))$ is a three-dimensional ancient $\kappa$-solution. Now we invoke the structure theorem for three-dimensional ancient $\kappa$-solutions in the precise form needed here: for the chosen $\varepsilon>0$, $A>0$, and integer $k\geq10$, every positive-curvature base point of a three-dimensional ancient $\kappa$-solution, after normalizing the scalar curvature at that point to $1$, lies in one of the standard $(\varepsilon,A,k)$ canonical-neighbourhood models. These models are the centre of a strong $\varepsilon$-neck, an $\varepsilon$-cap with cylindrical end, a compact positively curved component, or a pointed parabolic region controlled in $C^k$ by an ancient $\kappa$-solution model. At $(x_\infty,0)$ the normalization is already in force because the scalar curvature equals $1$. The final step is stability under convergence. Pointed $C^\infty_{\mathrm{loc}}$ convergence implies pointed $C^k$ convergence on the finite parabolic neighbourhood used in the $(\varepsilon,A,k)$ definition. Since each canonical-neighbourhood alternative is defined by $C^k$ closeness to a model after scaling, sufficiently close approximating flows have the same alternative. Hence, for all sufficiently large $i$, the point $(x_i,0)$ in $(M,\widetilde g_i(s))$ has an $(\varepsilon,A,k)$-canonical neighbourhood. Undoing the rescaling changes the model scale from $1$ back to $Q_i^{-1/2}$, so $(x_i,t_i)$ has the forbidden canonical neighbourhood in the original flow. This contradiction proves the existence of the curvature threshold.[/guided]

[step:Choose the curvature threshold and transfer the derivative estimates]The contradiction proves that there is a threshold \begin{align*} r=r(\varepsilon,A,k,T,g(0),v_0)>0 \end{align*} such that every point with $R(x,t)\geq r^{-2}$ has the asserted canonical neighbourhood at scale $R(x,t)^{-1/2}$. It remains to record the derivative bounds. By the compactness and derivative estimates for normalized ancient $\kappa$-solutions, for the fixed $\varepsilon$, $A$, $k$, and $\kappa$, there are constants $D_1=D_1(\varepsilon,A,k,\kappa)<\infty$ and $D_2=D_2(\varepsilon,A,k,\kappa)<\infty$ such that every normalized base point in every model appearing in the theorem satisfies \begin{align*} |\nabla R|\leq D_1 \end{align*} and \begin{align*} |\partial_t R|\leq D_2. \end{align*} This covers all alternatives: the shrinking-cylinder necks and caps have these bounds by the explicit model estimates, compact positively curved components are included in the compact part of the ancient $\kappa$-solution moduli after normalization, and the ancient-solution alternative is controlled by the same compactness statement. Equivalently, if one of these normalized quantities were unbounded, Hamilton compactness for normalized ancient $\kappa$-solutions would produce a smooth pointed ancient $\kappa$-solution limit with finite derivatives at its base point, contradicting smooth convergence. The $(\varepsilon,A,k)$-canonical-neighbourhood definition uses pointed parabolic $C^k$ closeness on the standard neighbourhood of spatial size $A$ in each listed alternative. Since $k\geq10$, this definition includes enough control of the metric and its time derivatives to compare the first spatial derivative and first time derivative of scalar curvature at the base point. Let $E=E(\varepsilon,A,k)<\infty$ be the comparison factor from this parabolic $C^k$ closeness, chosen so that the approximating normalized neighbourhood satisfies the two derivative bounds with $D_1$ and $D_2$ multiplied by $E$. Set \begin{align*} C_0:=E\max\{D_1,D_2\}. \end{align*} Because $\kappa=\kappa(g(0),T,v_0)$ is the finite-time noncollapsing constant determined by the initial metric and the initial unit-ball volume lower bound, $C_0$ depends only on $(\varepsilon,A,k,T,g(0),v_0)$. Under the parabolic rescaling $\widetilde g=R(x,t)g$ and $\widetilde s=R(x,t)(s-t)$, the scalar curvature satisfies $\widetilde R=R(x,t)^{-1}R$, while \begin{align*} |\widetilde\nabla \widetilde R|=R(x,t)^{-3/2}|\nabla R| \end{align*} and \begin{align*} |\partial_{\widetilde s}\widetilde R|=R(x,t)^{-2}|\partial_t R|. \end{align*} Thus, with \begin{align*} C=C(\varepsilon,A,k,T,g(0),v_0):=C_0<\infty, \end{align*} we obtain \begin{align*} |\nabla R|(x,t)\leq C R(x,t)^{3/2} \end{align*} and \begin{align*} |\partial_t R|(x,t)\leq C R(x,t)^2. \end{align*} This is the asserted scale-invariant derivative control and completes the proof.[/step]

[guided]The contradiction has already produced the threshold $r$. If no such threshold existed, the bad high-curvature sequence would have led to an ancient $\kappa$-solution limit and then back to a canonical neighbourhood at the bad points, which is impossible. Therefore there is \begin{align*} r=r(\varepsilon,A,k,T,g(0),v_0)>0 \end{align*} such that every point with $R(x,t)\geq r^{-2}$ has the required canonical neighbourhood at scale $R(x,t)^{-1/2}$. For the derivative estimates, we need constants that are uniform over all possible ancient $\kappa$-solution models, not constants chosen separately for each model. The compactness and derivative estimates for normalized ancient $\kappa$-solutions give exactly this. If no uniform bound for $|\nabla R|$ or $|\partial_tR|$ existed at normalized base points, we could take a sequence of normalized ancient $\kappa$-solutions with one of these quantities tending to infinity. Hamilton compactness would give a smooth pointed limit ancient $\kappa$-solution. Smooth convergence forces the corresponding derivatives at the base points to converge to finite derivatives of the limit, which contradicts unboundedness. Hence there are constants \begin{align*} D_1=D_1(\varepsilon,A,k,\kappa)<\infty \end{align*} and \begin{align*} D_2=D_2(\varepsilon,A,k,\kappa)<\infty \end{align*} such that every normalized model satisfies \begin{align*} |\nabla R|\leq D_1 \end{align*} and \begin{align*} |\partial_t R|\leq D_2. \end{align*} The $(\varepsilon,A,k)$-canonical-neighbourhood definition gives parabolic $C^k$ closeness to one of these models on the prescribed region of spatial size $A$. Since $k\geq10$, this closeness controls the first spatial derivative and the first time derivative of scalar curvature. Enlarging by a comparison factor $E=E(\varepsilon,A,k)<\infty$, define \begin{align*} C_0:=E\max\{D_1,D_2\}. \end{align*} The dependence is legitimate because $\kappa=\kappa(g(0),T,v_0)$ is supplied by finite-time noncollapsing from the initial metric and the initial unit-ball volume lower bound, so $C_0$ depends only on $(\varepsilon,A,k,T,g(0),v_0)$. Finally rescale back. If $\widetilde g=R(x,t)g$ and $\widetilde s=R(x,t)(s-t)$, then scalar curvature scales by \begin{align*} \widetilde R=R(x,t)^{-1}R. \end{align*} The spatial gradient of scalar curvature has scaling weight $3/2$, and the time derivative has scaling weight $2$: \begin{align*} |\widetilde\nabla \widetilde R|=R(x,t)^{-3/2}|\nabla R| \end{align*} and \begin{align*} |\partial_{\widetilde s}\widetilde R|=R(x,t)^{-2}|\partial_t R|. \end{align*} Since the normalized quantities are bounded by $C_0$, setting \begin{align*} C=C(\varepsilon,A,k,T,g(0),v_0):=C_0<\infty \end{align*} gives \begin{align*} |\nabla R|(x,t)\leq C R(x,t)^{3/2} \end{align*} and \begin{align*} |\partial_t R|(x,t)\leq C R(x,t)^2. \end{align*} This is the desired scale-invariant derivative estimate.[/guided]