Boundary as Clock: Time as Unified Translation Operator of Phase--Spectral Shift--Modular Flow

Against background of general C^\ast-algebras and operator scattering theory, construct framework of ``time = boundary translation.'' Time not viewed as pre-given flow parameter in bulk domain but defined as unique translation scale generated by boundary spectral data, maintaining self-consistency among phase--spectral shift--modular flow triple readings. Specifically: First, in scattering systems satisfying Birman--Krein conditions, take total scattering phase \Phi(\omega)=\arg\det S(\omega), s