[proofplan] The argument is a direct elimination of the alternatives supplied by the assumed [Canonical Neighborhood Theorem](/theorems/TEMP-65) conclusion at the high-curvature point $(x,t)$. The scalar-curvature lower bound places $(x,t)$ in the range where that conclusion applies. The epsilon-cap alternative must be handled with its internal structure: a point in an $\varepsilon$-cap is either in the cap core or in the neck part of the cap. Since the hypotheses exclude the cap-core and compact positively curved alternatives, the only remaining possibility is that a neighbourhood of $x$ at time $t$ is an epsilon-neck. [/proofplan] [step:Apply the canonical neighborhood alternatives at the high-curvature point] Let $Q := R(x,t)$ denote the scalar curvature at the point under consideration. The hypothesis $R(x,t) \geq r^{-2}$ gives $Q \geq r^{-2}$, so $(x,t)$ is one of the points to which the assumed [Canonical Neighborhood Theorem](/theorems/TEMP-65) conclusion with parameters $(\varepsilon,A,k)$ applies. In the formulation fixed in the statement, that conclusion gives exactly the following alternatives: $x$ is the center of an epsilon-neck; $x$ lies in an epsilon-cap; or $x$ lies in a compact positively curved component. In the cap alternative, the cap has a core by the convention fixed in the theorem statement, and every point of the cap outside that core is the center of an $\varepsilon$-neck at the same time. [guided] Define $Q := R(x,t)$, the scalar curvature of the Ricci flow $g(t)$ at the spacetime point $(x,t)$. The [Canonical Neighborhood Theorem](/theorems/TEMP-65) conclusion is assumed in the statement only for points whose scalar curvature is at least $r^{-2}$. The given lower bound \begin{align*} R(x,t) \geq r^{-2} \end{align*} therefore verifies the curvature-threshold hypothesis needed to invoke that conclusion at $(x,t)$. The statement fixes the precise form of the canonical-neighborhood conclusion used here. At the point $(x,t)$ there are three alternatives: $x$ is the center of an epsilon-neck; $x$ lies in an epsilon-cap; or $x$ lies in a compact positively curved component. The cap alternative has internal structure: an $\varepsilon$-cap consists of a compact core together with a neck part, and every point of the cap outside the core is the center of an $\varepsilon$-neck at the same time. Now use the two exclusion hypotheses. First, the theorem assumes that $x$ is not contained in the core of an $\varepsilon$-cap. Therefore, if the canonical-neighborhood alternative at $x$ is an $\varepsilon$-cap, then $x$ cannot be in the core and must be in the non-core neck part of that cap. By the cap convention just stated, this already gives an $\varepsilon$-neck neighbourhood of $x$ at time $t$. Second, the theorem assumes that $x$ is not in a compact positively curved component, so the compact positively curved component alternative is impossible. Hence every remaining canonical-neighborhood alternative gives an $\varepsilon$-neck neighbourhood of $x$ at time $t$. This is exactly the asserted conclusion. [/guided] [/step] [step:Eliminate the cap-core and compact-component alternatives] The theorem assumes that $x$ is not contained in the core of an epsilon-cap. If the canonical-neighborhood alternative at $x$ were an $\varepsilon$-cap, then either $x$ would lie in the core, which is excluded by hypothesis, or $x$ would lie in the non-core neck part of the cap, which by the cap convention is already an epsilon-neck neighbourhood. The theorem also assumes that $x$ is not in a compact positively curved component, so the compact positively curved component alternative is excluded. [/step] [step:Conclude that the remaining local model is an $\varepsilon$-neck] After the preceding exclusions, every possible canonical-neighborhood alternative gives an epsilon-neck neighbourhood of $x$ at time $t$: either the neck alternative holds directly, or $x$ lies in the non-core neck part of an epsilon-cap. By the definition of an $\varepsilon$-neck, this means that a neighbourhood of $x$ in the time-$t$ Riemannian manifold $(M,g(t))$ is an $\varepsilon$-neck. This is precisely the asserted conclusion. [/step]