Unified Time Scale and Boundary Time Geometry:\\ Single Structural Framework of Scattering Phase, Modular Flow, and Gravitational Boundary Terms

We construct a time unification framework with boundary as fundamental stage, integrating three originally separate time structures into different projections of the same ``boundary time geometry'': (1) On scattering and spectral theory end, based on Birman--Kreĭn formula and Wigner--Smith time delay, prove scale identity among total scattering phase derivative, relative state density, and group delay trace; (2) On operator algebra and information end, based on Tomita--Takesaki modular theory and Connes--Rovelli thermal time hypothesis, characterize modular flow parameter as intrinsic time determined by state--algebra pair, introducing time scale equivalence class; (3) On gravity and geometry end, based on Einstein--Hilbert--Gibbons--Hawking--York action and its boundary variation, unify extrinsic curvature and time translation generated by boundary Hamiltonian into same boundary time geometry. In unified model, boundary is described by triple structure: intrinsic metric and extrinsic curvature of geometric boundary \partial M, quantum boundary algebra A_\partial with state \omega, and scattering matrix S(\omega) defined in external region. Under well-posed traceable scattering assumptions, construct ``scale identity'' $ \varphi'(\omega){\pi}=\rho_{rel}(\omega)=1{2\pi}trQ(\omega), where \varphi(\omega)=1{2}\arg\det S(\omega) is total scattering phase, \rho_{rel} is derivative of spectral shift density, Q(\omega)=-iS(\omega)^\dagger\partial_\omega S(\omega) is Wigner--Smith time delay matrix. This scale is standardized at modular time and geometric time ends respectively through modular Hamiltonian operator K_\omega=-\log\Delta_\omega and Hamilton--Jacobi functional of GHY boundary action, thus defining single boundary time scale equivalence class [\tau]. Based on this, we give several unification theorems: (i) Categorical existence uniqueness of time scale equivalence class: on common domain of given boundary algebra, scattering data, and gravitational boundary geometry, all acceptable time parameters are monotonic rescalings of one fundamental boundary time; (ii) Cosmological redshift relation 1+z=1/a(t)$ can be interpreted as global rescaling of this time equivalence class on large scales, thus unifying local scattering time delay with conformal time in FRW background; (iii) In cases with horizons (Rindler wedge and black hole exterior), modular flow time, proper time of accelerated observer, and geometric outward normal translation time fall into same equivalence class. Full text gives explicit assumptions and theorems at scattering--spectral, modular flow--information, and gravity--boundary geometry ends respectively, with detailed proofs of scale identity, modular time equivalence, and boundary Hamiltonian generated time in appendices, finally proposing engineered measurement schemes based on waveguides, microwave cavities, and Aharonov--Bohm rings to cross-calibrate three types of time scales experimentally.