Optimal Extension Regularity at the McVittie Event Horizon

We determine the optimal local extension regularity of the future black-hole event horizon in the exact spatially flat McVittie solutions sourced by a positive cosmological constant and a barotropic fluid with constant equation-of-state parameter $w>-1$. Let $H_{\mathrm{\infty }}$ be the asymptotic Hubble constant, $\kappa$ the surface gravity of the limiting black-hole root, and $p=3(1+w)H_{\mathrm{\infty }}\mathrm{/}\kappa$. Ingoing radial null geodesics reach the horizon in finite affine length. A parallelly propagated angular curvature component is asymptotic to $C{s}^{p-2}$, with $C\mathrm{\ne }0$ and $s$ the remaining affine distance, which excludes every anchored $C^2$ extension for $0<p<2$. For $p\ge 2$ we construct a parameter-uniform Gaussian-null compactification and an explicit two-sided Lorentzian collar. If $p=N+\vartheta$ is nonintegral, with $N\ge 2$ and $0<\vartheta <1$, the optimal regularity is the standard big Hölder class $C^{N,\vartheta }$: extensions of this class exist, whereas no $C^{N,{\vartheta }^{\mathrm{\prime }}}$ extension exists for ${\vartheta }^{\mathrm{\prime }}>\vartheta$. Every integer $p\ge 2$ instead belongs to an analytic island and admits a real-analytic local extension. At the critical value $p=2$ the boundary Einstein endomorphism has a nonzero rank-one nilpotent part. The ratio of cosmological decay to horizon redshift therefore determines a sharp, arithmetic hierarchy of geometric regularity.