[proofplan] We treat the statement as a synthesis theorem with explicitly stated regimes: the conclusion is obtained by applying the standard structural inputs for three-dimensional $\kappa$-solutions rather than by deriving Perelman's canonical-neighbourhood theorem from scratch. First we unpack the hypotheses in the definition of a $\kappa$-solution and record that they are precisely the hypotheses needed for compactness, pinching, canonical-neighbourhood, and asymptotic-soliton arguments. Then we apply the canonical-neighbourhood alternative to high-curvature pointed regions, keeping the compact spherical-type alternative separate from the neck-and-cap alternatives and recording the dependence of the curvature threshold. We use Perelman's reduced-distance blow-down and asymptotic-neck theorems only in the noncompact regimes where their hypotheses give cylindrical limits, and finally explain why the compact conclusion is a canonical-neighbourhood and conditional normalized-limit statement rather than an exact classification of compact ancient solutions. [/proofplan] [step:Unpack the $\kappa$-solution hypotheses used by the structural theorems] By definition, a three-dimensional $\kappa$-solution is a complete ancient Ricci flow $(M^3,g(t))_{t \in (-\infty,0]}$ with bounded curvature on each time slice, nonnegative curvature operator, positive scalar curvature, and $\kappa$-noncollapsing on all scales. The positivity of scalar curvature is global in the definition used here; equivalently, for a nonflat ancient solution with nonnegative curvature operator it is supplied by the strong maximum principle for the scalar-curvature evolution. These are exactly the inputs used by [Hamilton-Ivey pinching](/page/Hamilton-Ivey%20Pinching), [Hamilton compactness for pointed Ricci flows](/page/Hamilton%20Compactness%20Theorem), [Perelman's canonical-neighbourhood theorem](/page/Canonical%20Neighbourhood%20Theorem) for three-dimensional $\kappa$-solutions, and [Perelman's asymptotic-shrinker theorem](/page/Asymptotic%20Shrinker%20Theorem) for noncompact ancient solutions. Fix $\varepsilon>0$. In this proof, an $\varepsilon$-neck at a spacetime point means the standard parabolic neighbourhood used in Perelman's three-dimensional canonical-neighbourhood theorem: after scaling by the scalar curvature at the basepoint and shifting the base time to $0$, the pointed flow on the theorem's normalized spacetime cylinder is $\varepsilon$-close in pointed smooth Cheeger-Gromov topology, including all derivatives up to the order specified in that theorem, to the corresponding normalized region of the shrinking round cylinder $S^2\times\mathbb{R}$ or of a finite isometric quotient. An $\varepsilon$-cap means one of the cap models in the same theorem: a connected high-curvature region diffeomorphic to a three-ball or a one-sided finite quotient cap whose boundary side is an $\varepsilon$-neck in this parabolic sense. A canonical-neighbourhood regime means that the basepoint scalar curvature is above the threshold supplied by Perelman's canonical-neighbourhood theorem for this fixed $\varepsilon$, $\kappa$, dimension $3$, and the chosen scalar-curvature normalization. For a spacetime point $(x,t) \in M \times (-\infty,0]$ with scalar curvature $R(x,t)>0$, define the parabolically rescaled metric family $g_{(x,t)}: (-\infty,0] \to \Gamma(S^2T^*M)$ by \begin{align*} g_{(x,t)}(s) := R(x,t)\, g\left(t + \frac{s}{R(x,t)}\right). \end{align*} Here $\Gamma(S^2T^*M)$ denotes the space of smooth symmetric covariant two-tensor fields on $M$. We write the resulting pointed Ricci flow as $(M,g_{(x,t)}(s),x)_{s \le 0}$. This rescaling normalizes the scalar curvature at the basepoint to $1$ and preserves completeness, nonnegative curvature operator, and $\kappa$-noncollapsing, while converting high-curvature local questions into bounded-curvature pointed compactness questions. [/step] [step:Apply the canonical-neighbourhood alternative at the curvature scale] Perelman's [canonical-neighbourhood theorem](/page/Canonical%20Neighbourhood%20Theorem), in the three-dimensional scalar-curvature normalization fixed above, says that for the fixed parameters $\varepsilon>0$ and $\kappa>0$ there is a curvature threshold $Q_{\varepsilon,\kappa}<\infty$, depending only on $\varepsilon$, $\kappa$, dimension $3$, and that normalization convention, such that every spacetime point $(x,t)$ of a three-dimensional $\kappa$-solution with $R(x,t)\ge Q_{\varepsilon,\kappa}$ has, after the rescaling $g_{(x,t)}$, one of the standard canonical neighbourhoods at unit scalar-curvature scale. The hypotheses are exactly those verified above: completeness, ancientness, bounded curvature on time slices, nonnegative curvature operator, positive scalar curvature, and $\kappa$-noncollapsing on all scales. In the coarse three-dimensional formulation used here, the alternatives are: an $\varepsilon$-neck modeled on the shrinking round cylinder $S^2\times\mathbb{R}$ or a finite isometric quotient of it; an $\varepsilon$-cap whose boundary side is such a neck; or, in the compact positively curved case, a spherical-type canonical model. Thus the neck-and-cap assertion applies precisely in the non-spherical canonical-neighbourhood regimes, while the compact spherical alternative is handled separately below. [guided] Fix $\varepsilon>0$. The precise input is Perelman's [canonical-neighbourhood theorem](/page/Canonical%20Neighbourhood%20Theorem): for this $\varepsilon$ and for the noncollapsing constant $\kappa$, there exists a number $Q_{\varepsilon,\kappa}<\infty$ such that if $(x,t)$ is a spacetime point in a three-dimensional $\kappa$-solution and $R(x,t)\ge Q_{\varepsilon,\kappa}$, then the geometry around $(x,t)$ at the curvature scale $R(x,t)^{-1/2}$ is one of the standard canonical models: a cylindrical neck, a cap attached to a cylindrical neck, or a compact spherical-type model in the positively curved compact alternative. This statement is made after the parabolic normalization \begin{align*} (M,g_{(x,t)}(s),x)_{s \le 0}, \qquad g_{(x,t)}(s) := R(x,t)\, g\left(t + \frac{s}{R(x,t)}\right), \end{align*} which makes the scalar curvature at the basepoint equal to $1$ at time $s=0$. We verify the theorem's hypotheses. The original flow is complete and ancient by the definition of a $\kappa$-solution. Its curvature operator is nonnegative, its scalar curvature is positive, and its curvature is bounded on each time slice, again by definition. The $\kappa$-noncollapsing condition is scale-invariant, so it is still available after replacing $g(t)$ by $g_{(x,t)}(s)$. Therefore the canonical-neighbourhood theorem applies to every point with $R(x,t)\ge Q_{\varepsilon,\kappa}$. The conclusion is not merely that some compactness subsequence exists. It is the stronger local alternative at the basepoint scale: outside the compact spherical-type alternative, the normalized parabolic neighbourhood is $\varepsilon$-close in pointed smooth Cheeger-Gromov topology, on the normalized parabolic neighbourhood and derivative order specified by the theorem, to a shrinking round cylindrical model $S^2\times\mathbb{R}$, or to a finite isometric quotient of that model, unless the point lies in an $\varepsilon$-cap whose boundary side is such a neck. If the compact spherical-type alternative occurs, it is not a neck or cap assertion; it belongs to the compact limiting discussion below. This is exactly the high-curvature canonical-neighbourhood assertion used later in the course. [/guided] [/step] [step:Realize the noncompact blow-down through the asymptotic shrinker] Assume now that $M$ is noncompact. Fix a base spacetime point $(p,0) \in M \times \{0\}$. The reduced distance based at $(p,0)$ is the function $\ell_p: M \times (-\infty,0) \to [0,\infty)$ appearing in Perelman's reduced-volume monotonicity formula, and a controlled reduced-distance region means a sequence of spacetime points $(x_j,t_j)$ with $t_j\to -\infty$ and $\sup_j \ell_p(x_j,t_j)<\infty$. The associated blow-down sequence is the pointed Ricci flow obtained by the parabolic dilation. For each $j$, define the metric family $g_j: (-\infty,0] \to \Gamma(S^2T^*M)$ by \begin{align*} g_j(s) := |t_j|^{-1} g(t_j + |t_j|s). \end{align*} The flow is pointed at $x_j$. This large-time blow-down is different from the basepoint scalar-curvature normalization $g_{(x,t)}$ above, which is a curvature-scale blow-up used for canonical neighbourhoods. Perelman's [asymptotic-shrinker theorem](/page/Asymptotic%20Shrinker%20Theorem) supplies the compactness estimates for this construction: on each compact parabolic neighbourhood of the selected controlled reduced-distance basepoints, the rescaled flows have uniform curvature bounds and a uniform injectivity-radius lower bound from $\kappa$-noncollapsing. Thus [Hamilton compactness](/page/Hamilton%20Compactness%20Theorem) gives a smooth pointed subsequential limit, and the monotonicity of reduced volume identifies that limit as a nonflat gradient shrinking soliton. In the noncompact course regime, the final structural input is the noncompact conclusion in Perelman's [asymptotic-shrinker theorem](/page/Asymptotic%20Shrinker%20Theorem) and its asymptotic-neck refinement: for controlled reduced-distance blow-downs and for sequences escaping to spatial infinity to which the asymptotic-neck theorem applies, the limiting shrinker is noncompact and cylindrical. Once this noncompactness conclusion is supplied by that theorem, the [classification of three-dimensional nonflat noncompact shrinking solitons](/page/Classification%20of%20Three-Dimensional%20Shrinking%20Solitons) leaves only the shrinking round cylinder $S^2\times\mathbb{R}$ or a finite isometric quotient under nonnegative curvature operator and noncollapsing. Therefore the cylindrical conclusion holds exactly for the blow-down-at-infinity sequences covered by the reduced-distance asymptotic-shrinker construction and for the corresponding escaping asymptotic-neck subsequences used in the course. [guided] The point of this step is to distinguish two normalizations. The canonical-neighbourhood normalization rescales by the scalar curvature at a chosen point. A blow-down at infinity instead rescales by a large backward time scale. Fix a basepoint $(p,0)$ and let $\ell_p: M \times (-\infty,0) \to [0,\infty)$ be Perelman's reduced-distance function based there. If $t_j\to -\infty$ and $\sup_j \ell_p(x_j,t_j)<\infty$, define the metric family $g_j: (-\infty,0] \to \Gamma(S^2T^*M)$ by \begin{align*} g_j(s) := |t_j|^{-1} g(t_j + |t_j|s). \end{align*} The pointed flows $(M,g_j(s),x_j)$ are the reduced-distance blow-down sequence. We apply Perelman's [asymptotic-shrinker theorem](/page/Asymptotic%20Shrinker%20Theorem). Its hypotheses are the three-dimensional $\kappa$-solution hypotheses already verified: ancientness, completeness, bounded curvature on time slices, nonnegative curvature operator, positive scalar curvature, and $\kappa$-noncollapsing. The controlled reduced-distance condition supplies the basepoints for which the reduced-volume compactness argument gives uniform curvature bounds on compact parabolic neighbourhoods. The $\kappa$-noncollapsing hypothesis gives the corresponding injectivity-radius lower bounds. Therefore [Hamilton compactness](/page/Hamilton%20Compactness%20Theorem) applies and gives a smooth pointed subsequential limit. Reduced-volume monotonicity identifies that limit as a nonflat gradient shrinking soliton. The noncompactness of the limiting shrinker is not inferred merely from the noncompactness of $M$. It is part of the noncompact asymptotic-shrinker and asymptotic-neck input used in this course: in the controlled reduced-distance blow-down regime, and in the escaping spatial-infinity regime covered by the asymptotic-neck theorem, the subsequential limit is a noncompact shrinking model. The [classification of three-dimensional nonflat noncompact shrinking solitons](/page/Classification%20of%20Three-Dimensional%20Shrinking%20Solitons) then leaves only the shrinking round cylinder $S^2\times\mathbb{R}$ and its finite isometric quotients. The asymptotic-neck formulation gives the same conclusion for sequences of spatial points escaping to infinity after passing to subsequences at the curvature scale. This proves the noncompact cylindrical blow-down assertion in the precise sense used by the course. [/guided] [/step] [step:Interpret the compact case as canonical-neighbourhood and conditional spherical limiting behaviour] Assume instead that $M$ is compact. The same [canonical-neighbourhood theorem](/page/Canonical%20Neighbourhood%20Theorem) applies at every spacetime point with $R(x,t)\ge Q_{\varepsilon,\kappa}$, so compact high-curvature regions are covered by the standard canonical-neighbourhood alternatives: necks, caps, and the compact spherical-type alternative. Thus the compact part is not exhausted by the neck-and-cap language alone. The spherical assertion is only a normalized-limit statement, exactly as stated in the theorem. Precisely, suppose a sequence of compact three-dimensional $\kappa$-solutions is normalized so that Hamilton compactness applies, and suppose the course regime supplies one of the two additional hypotheses named in the statement: either curvature-operator pinching to constant positive sectional curvature or convergence to a compact shrinking-soliton limit. Under the first hypothesis, the curvature-operator pinching is preserved under smooth Cheeger-Gromov convergence, so the Hamilton limit has constant positive sectional curvature. Under the second hypothesis, the compact three-dimensional shrinking-soliton classification gives a quotient of the shrinking round three-sphere by a finite group of isometries acting freely. In either case the limiting compact model is a spherical space form. This proves only the compact canonical-neighbourhood and conditional normalized-limit conclusions stated here; it does not assert that every compact ancient three-dimensional $\kappa$-solution is exactly a round spherical space form at every time. [/step] [step:Combine the structural alternatives] The high-curvature canonical-neighbourhood alternative gives cylindrical necks, finite cylindrical quotients, caps attached to necks, and the compact spherical-type alternative at the scalar-curvature scale. The noncompact asymptotic-shrinker and asymptotic-neck inputs identify the models at infinity as cylindrical for the reduced-distance blow-down sequences and the corresponding escaping subsequences used in the course. The compact discussion gives the corresponding canonical-neighbourhood and conditional normalized-limit spherical behaviour without upgrading it to an exact classification of all compact ancient $\kappa$-solutions. These conclusions are exactly the course-level structure asserted in the theorem. [/step]