Universal Conservation of Information Rate: From Quantum Cellular Automata to Unified Framework of Relativity, Mass, and Gravity

Within quantum cellular automaton (QCA) and finite information ontology framework, construct effective description of single-particle long-wavelength excitations; prove core result based on Hilbert space geometry and unitarity: for any discrete quantum walk/QCA defined by local unitary evolution and translation invariance, emerging one-dimensional Dirac-type Hamiltonian in continuum limit, long-wavelength single-particle eigenmode external group velocity (v_{ext}) and internal state evolution ve