Windowed Formulation of Phase--Spectral-Shift--DOS--Cosmological-Constant

We establish equivalence chain centered on generalized scattering phase, connecting Kontsevich--Vishik (KV) determinant phase, generalized Kreĭn spectral shift, density-of-states difference (DOS difference), and Wigner--Smith (WS) trace identity, providing rigorous variable and factor accounting under even-dimensional asymptotically hyperbolic/conformally compact (AH/CCM) and static-patch de Sitter (dS) geometry. Under strict Lifshits--Kreĭn (LK) trace formula and relative trace-class assumption