Boundary as Unified Stage: Variational Completeness, Time Scale, Topological Branching

From first principles, elevate ``boundary'' from passive geometric appendage to unified physical stage. Propose axiomatic framework with boundary as fundamental object, gluing three seemingly separate structures—Gibbons--Hawking--York (GHY) boundary term and Brown--York quasilocal quantities in gravitational variation, spectral shift function and Wigner--Smith time delay in scattering theory, modular flow and relative entropy monotonicity in operator algebras—as different projections of same ``b