Positive $\tau$-bi-Ricci curvature and Mean curvature flow with surgery in hyperbolic space

Let $n\ge 3$ and $0\le \tau \le 2$. We prove that every smooth closed connected immersed hypersurface in hyperbolic space whose induced metric has positive $\tau$-bi-Ricci curvature admits a mean curvature flow with surgery which has only finitely many surgery times and terminates. In the range compatible with cylindrical necks, the key ingredients are a preserved quantitative spectral pinching condition, which converts the intrinsic hypothesis into uniform two-convexity, and cylindrical and derivative estimates that remain valid across the hyperbolic standard neck replacement. At the endpoint $n=3$, $\tau =2$, positive $\tau$-bi-Ricci curvature is positive Ricci curvature and forces strict convexity, so the ordinary mean curvature flow converges to a round point. Consequently, the underlying manifold is diffeomorphic to a sphere or to a finite connected sum of copies of ${\mathbb{S}}^{n-1}\times {\mathbb{S}}^1$. If the initial hypersurface is embedded and bounds a compact domain, that domain is a one-handlebody, namely a ball with finitely many one-handles attached.