Geometry-induced Casimir response in a helicoidal spacetime

We investigate the Casimir response of a massive scalar field in a torsionless helicoidal spacetime with Levi-Civita connection. The background is ultrastatic and curved, with scalar curvature $R=-2\Omega^2$, and its off-diagonal metric component induces a geometric coupling between angular and axial quantum numbers. We show that individual modes exhibit a linear helicoidal splitting, whereas the linear contribution cancels in the nonchiral vacuum sum. The leading Casimir correction is therefore quadratic in the twist and defines a helicoidal vacuum susceptibility after local ultraviolet subtractions. For a cylindrical Dirichlet cavity, we compute this scheme-defined susceptibility and the associated correction to the radial Casimir force. The results identify torsionless helicoidal geometry as a controlled setting in which mode-level chirality produces a finite quadratic response of vacuum fluctuations.