Quantum--Classical Bridge on Time Scale:\\ Phase, Time Delay, Redshift, and Gravity--Entropy Geometry

Under the unified ``time scale'' perspective, this paper systematically constructs a set of equivalence relations among quantum phase, proper time, scattering group delay, cosmological redshift, and boundary entropy evolution, organizing them into an axiomatizable quantum--classical bridge framework. On the geometric end, proper time d\tau=-g_{\mu\nudx^\mu dx^\nu} and generalized entropy S_{gen} on causal boundaries are taken as fundamental objects; on the quantum end, path integral phase \phi=-S/\hbar, total phase \Phi(\omega) of scattering matrix S(\omega), and Wigner--Smith group delay operator Q(\omega)=-iS(\omega)^\dagger \partial_\omega S(\omega) are taken as fundamental objects. We propose and adopt the unified scale identity $ \varphi'(\omega){\pi}=\rho_{rel}(\omega)=1{2\pi}trQ(\omega), where \varphi(\omega) is normalized total phase, \rho_{rel} is relative state density. We prove: under semiclassical limit and appropriate regularity assumptions enumerate \item When single-particle wave packet propagates along classical geodesic, its phase \phi is equivalent to a linear function of proper time accumulated along that worldline \phi=-mc^2\int d\tau/\hbar, thus proper time scale can be viewed as ``geometric parameter of phase''; \item In static or asymptotically flat gravitational fields, gravitational time delay of photons or matter waves equals derivative of scattering phase with respect to frequency, i.e., equals trace of Wigner--Smith group delay, thus gravitational time dilation can be interpreted as curvature of ``phase--frequency geometry''; \item In FRW cosmological background, redshift 1+z=a(t_0)/a(t_e) can be equivalently written as ratio of phase frequency (d\phi/dt) on same photon worldline at emission and detection events, thus cosmological redshift becomes phase expression of ``cosmic time scale shear''; \item In local causal diamonds, taking extremality and monotonicity of generalized entropy S_{gen}=A/(4G\hbar)+S_{out} (along null generator parameter \lambda$) as axioms, Einstein equations with cosmological constant can be derived in semiclassical--holographic window, viewing gravitational geometry as ``effective equations for how entropy organizes along time scale on causal boundaries.'' enumerate This yields a unified ``time--phase--entropy--geometry'' correspondence diagram. Appendices provide: derivation of scale identity between Wigner--Smith group delay and spectral shift--state density; semiclassical proof of ``phase--proper time'' equivalence under worldline path integral; refinement of redshift--phase expression in FRW cosmology; and detailed proof outline of deriving Einstein equations from generalized entropy extremality and quantum energy conditions on local causal diamonds.