Fix $\varepsilon>0$, $A>0$, an integer $k\geq 10$, and $T<\infty$. Let $(M^3,g(t))$, $t \in [0,T)$, be a Ricci flow on a closed three-manifold. Suppose the initial metric is normalized by

\begin{align*} |\operatorname{Rm}|_{g(0)}\leq 1 \end{align*}

and that every unit ball in $(M,g(0))$ has volume at least $v_0>0$. Then there exist constants $r=r(\varepsilon,A,k,T,g(0),v_0)>0$ and $C=C(\varepsilon,A,k,T,g(0),v_0)<\infty$ such that if $R(x,t) \geq r^{-2}$, then $(x,t)$ has an $(\varepsilon,A,k)$-canonical neighbourhood at scale $R(x,t)^{-1/2}$. In particular, after parabolic rescaling by $R(x,t)$, the point lies in one of the following alternatives on the standard backward parabolic neighbourhood of spatial size $A$: the centre of a strong $\varepsilon$-neck, an $\varepsilon$-cap whose end is a strong $\varepsilon$-neck, a compact positively curved component, or a region $C^k$-close to a three-dimensional ancient $\kappa$-solution with nonnegative curvature operator, where $\kappa=\kappa(g(0),T)>0$ is the finite-time noncollapsing constant. The $C^k$ parabolic closeness in this definition includes the derivatives needed to control $\nabla R$ and $\partial_tR$. In every case the curvature derivatives satisfy scale-invariant estimates, for example

\begin{align*} |\nabla R|(x,t) \leq C R(x,t)^{3/2}, \qquad |\partial_t R|(x,t) \leq C R(x,t)^2. \end{align*}