Unified Framework of Time--Geometry--Interaction:\\ From Scattering Phase Scale to Geometrization of Gravity and Gauge Forces

We construct a unified framework with ``time scale equivalence class'' as the core object, rewriting gravity, gauge interactions, and macroscopic classical ``forces'' as projections of the same geometric structure at different hierarchical levels. First, in the rigorous context of scattering theory, based on Birman--Kreĭn spectral shift function and Wigner--Smith time delay, we give the scale identity of phase derivative, relative state density, and group delay trace: for a class of scattering systems satisfying trace-class perturbation conditions, we have \varphi'(\omega)/\pi=\rho_{rel}(\omega)=(2\pi)^{-1}tr\,Q(\omega), where Q(\omega)=-iS(\omega)^{\dagger}\partial_{\omega}S(\omega) is the Wigner--Smith matrix, \rho_{rel} is the derivative of Kreĭn spectral shift density. This identity unifies ``phase--delay--relative spectral density'' as an observable time scale. Second, introducing a total bundle with spacetime as base manifold and internal charge space and ``observer resolution hierarchy'' as fibers, we prove that under conditions satisfying local Lorentz and internal gauge symmetry, gravity and Yang--Mills gauge fields can be viewed as different components of the same total connection, whose curvatures respectively give spacetime curvature and internal field strength; the time scale is related to phase integration along worldlines and parallel transport of the connection, thus deriving the unified proposition that ``no fundamental forces, only time curvature under different geometric projections.'' Third, at the algebraic quantum field theory level, using modular flow given by Tomita--Takesaki modular theory as ``internal time,'' combined with quantum null energy condition (QNEC) and generalized entropy monotonicity, we characterize the arrow of time as a partial order structure of relative entropy and modular energy monotonicity. Finally, in the semiclassical limit, we give a theorem-form statement of ``force = projected curvature of time geometry'' for expectation value evolution of macroscopic particles, explaining that classical Newtonian mechanics, Lorentz force, and even effective entropic force can all be viewed as effective descriptions of this unified time geometry under different coarse-graining and internal state constraints. At the end, we provide several verifiable engineering proposals, including Wigner--Smith time delay measurements in microwave networks and mesoscopic conductors, atomic clock gravitational redshift--phase scale reconciliation, and cross-scale scale consistency tests based on FRW cosmological redshift and strong gravitational lensing.