Unified Time Scale and Time Geometry:\\ Causal Ordering, Unitary Evolution, and Generalized Entropy

We propose a ``unified time scale equivalence class'' rigorously gluing three ends of relativity, quantum scattering, and information--holography. Core scale identity unifies derivative of total scattering phase, relative state density, and trace of Wigner--Smith group delay as different projections of same object: $ \ \frac{\varphi'(\omega){\pi}\;=\;\rho_{rel}(\omega)\;=\;1{2\pi}trQ(\omega)\ },\qquad Q(\omega)=-\,iS(\omega)^\dagger\partial_\omega S(\omega),\ \varphi=1{2}\arg\det S. In geometry end, Killing time, ADM lapse, null geodesic affine parameter, and FRW conformal time proven mutually rescalable within unified equivalence class; in information--holography end, taking Tomita--Takesaki modular flow as ``intrinsic time,'' controlling generalized entropy extremality on small causal diamonds with QFC/QNEC and relative entropy monotonicity, thus deriving Einstein equations in semiclassical--holographic window. Framework obtains three alignments: (i) phase--proper time equivalence \phi=(mc^2/\hbar)\int d\tau; (ii) gravitational time delay = group delay trace \Delta T=\partial_\omega\Phi=TrQ; (iii) FRW redshift = phase rhythm ratio 1+z=a(t_0)/a(t_e)=[(d\phi/dt)_e]/[(d\phi/dt)_0]$. This paper under three axioms of causal ordering--unitary evolution--entropy monotonicity/extremality, establishes existence and affine uniqueness of unified time scale, giving realization schemes for experimental and engineering metrology.