Dirac Operators, APS Boundary Conditions, and Spectral Flow on a Finite Warped Cylinder

We study the Dirac operator on a finite warped cylinder coupled to a background $U(1)$ gauge field. We identify the intrinsic endpoint operators defining the Atiyah-Patodi-Singer (APS) boundary condition and derive a determinant characterization of the modewise APS spectrum. In the constant-gauge, invertible setting, the endpoint reduced $\eta$ contributions cancel, so the APS index vanishes. For smooth gauge families, the APS projector becomes discontinuous when a boundary mode crosses zero. We therefore introduce a regularized APS-type family of self-adjoint endpoint conditions that remains continuous across such crossings. This regularized family admits a real-symplectic boundary formulation within the standard spectral-flow/Maslov framework: for nondegenerate regularization, the zero-mode set coincides with the boundary-zero set, and transverse boundary zeros give isolated regular crossings.