Unified Time Scale and Time Geometry:\\ Equivalence, Domains, and Solvable Models of Spectral--Scattering--Causal--Entropy

We propose and rigorously characterize a ``unified time scale'' framework, aligning phase gradient readings, relative state density, and trace of Wigner--Smith group delay within strict scattering theory domain, thus defining time scale as monotonic reparametrization of a class of spectral--scattering invariants. The identity $ \ \frac{\varphi'(\omega){\pi}=\rho_{rel}(\omega)=1{2\pi}TrQ(\omega)\ },\qquad Q(\omega)=-\,i\,S(\omega)^\dagger\partial_\omega S(\omega),\quad \varphi=1{2}\arg\det S holds within energy windows satisfying elastic--unitary scattering and Birman--Kreĭn assumptions; for absorptive/non-unitary and long-range potential cases, we propose verifiable generalizations: introducing complex time delay, dwell time, and phase renormalization, using Poisson--convolution to give existence and affine uniqueness of windowed clocks. Paper further constructs model-based proof of eikonal phase derivative = geometric Shapiro delay in general relativity end (Schwarzschild exterior scalar wave, high-frequency/high-angular-momentum limit), expresses redshift as phase rhythm ratio in cosmological end, and states ``entropy extremality \to$ geometric equations'' as conditional proposition in information--holography end with relative entropy monotonicity and QNEC as core assumptions. Entire text emphasizes domain of equivalence relations and solvable examples, giving engineering-realizable multi-frequency group delay metrology and lensing delay inversion schemes.