Boundary as Unified Stage: From Time Translation Operator, Null--Modular Double Cover to GHY Boundary Term

Construct unified framework with boundary as sole fundamental stage, gluing three mature but usually separate structures as three projections of same object: (i) Based on Birman--Krein--Friedel--Wigner--Smith framework, ``time translation operator'' characterized by scattering phase, spectral shift function, Wigner--Smith time delay matrix; (ii) Based on Tomita--Takesaki modular theory and recent results on causal diamond/null surface modular Hamiltonians, Markov property, QNEC, Null--Modular do