Time as Density of States: Unified Time Identity \kappa = \rho and Microscopic Scattering Mechanism of Gravitational Redshift

In conventional physics, time plays incompatible roles in quantum theory and general relativity: in quantum mechanics it is an external evolution parameter generated by a Hamiltonian, whereas in general relativity proper time is a path-dependent functional of the spacetime metric. Within a quantum cellular automaton (QCA) and optical-path conservation framework, this work proposes a microscopic definition of time in terms of the density of quantum states. Building on Eisenbud--Wigner--Smith (EWS