Sharp focal radius estimate and rigidity of hypersurfaces in manifolds with positive curvature

We prove a sharp Clifford-threshold focal-radius estimate and rigidity for immersed hypersurfaces. Under a $p$-form curvature condition, formulated by the Weitzenböck curvature term together with ${\mathrm{R}\mathrm{i}\mathrm{c}}_p\ge p$, any closed two-sided immersion $F:{\mathrm{\Sigma }}^m\to M^{m+1}$ with $b_p(\mathrm{\Sigma };\mathbb{R})\mathrm{\ne }0$ and $1\le p\le m\mathrm{/}2$ satisfies $$r_f(F,M)\le \frac{\pi }{4}\mathrm{.}$$ The equality case is rigid: if the ambient manifold is complete, equality forces the hypersurface to be locally the Clifford hypersurface $S^p(1\mathrm{/}\sqrt{2})\times S^{m-p}(1\mathrm{/}\sqrt{2})\subset S^{m+1}(1)$; if the ambient manifold is compact and connected, it is a spherical space form. The curvature condition follows from $\sec \ge 1$ for $p=1$, from normalized $\mathrm{P}\mathrm{I}\mathrm{C}1\ge 1$ for $p=2$, and from curvature operator bounded below by one in all degrees. By quotient lifting and the Hopf fibrations, we also obtain focal-radius estimates in ${\mathbb{C}\mathbb{P}}^n$ and ${\mathbb{H}\mathbb{P}}^n$, with projective Clifford rigidity, without any Betti-number assumption.