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

In the framework of Quantum Cellular Automata (QCA) and finite information ontology, we construct an effective description of single-particle long-wavelength excitations and prove a core result based on Hilbert space geometry and unitarity: for any discrete quantum walk/QCA defined by local unitary evolution and translation invariance that emerges a one-dimensional Dirac-type Hamiltonian in the continuous limit, the external group velocity (v_{ext}) and internal state evolution velocity (v_{int}