Let $g:(-\infty,0]\to\Gamma(S^2T^*M)$ be a complete nonflat ancient Ricci flow on a three-manifold $M$, and suppose $(M^3,g(t))_{t\le 0}$ is a three-dimensional $\kappa$-solution with bounded nonnegative curvature operator and positive scalar curvature. For every $\varepsilon>0$ there is a curvature threshold $Q_{\varepsilon,\kappa}<\infty$ such that every spacetime point $(x,t)$ with $R(x,t)\ge Q_{\varepsilon,\kappa}$ has an $\varepsilon$-canonical neighbourhood. More precisely, after rescaling by $R(x,t)$, the neighbourhood is $\varepsilon$-close in the prescribed high-$C^k$ topology to one of the standard models: a round shrinking cylindrical neck $S^2\times\mathbb R$ or a finite isometric quotient, an $\varepsilon$-cap attached to such a neck, or a compact positively curved component. In the noncompact case, the asymptotic necks obtained by escaping to infinity have cylindrical subsequential limits, again $S^2\times\mathbb R$ or finite isometric quotients.