Universe as Quantum Discrete Cellular Automaton: Axiomatic Characterization

Within the frameworks of Quantum Cellular Automata (QCA), quasi-local C^\ast-algebras, and causal sets, we construct an axiomatic system for "Universe = Single Quantum Discrete Cellular Automaton". Specifically, we use a countable connected graph \Lambda as the discrete space; a finite-dimensional local Hilbert space \mathcal H_{\rm cell} and the quasi-local algebra \mathcal A on its infinite tensor product to describe local quantum degrees of freedom; a \ast-automorphism \alpha:\mathcal A\to\mathcal A with finite propagation radius and its unitary implementation U to describe discrete time evolution; and an initial cosmic state \omega_0 to describe the universe at time n=0. We prove that under these axioms, the naturally induced relation on the event set E=\Lambda\times\mathbb Z constitutes a locally finite partial order, thereby yielding a discrete causal set whose local finiteness strictly corresponds to the finite propagation radius condition of the QCA, aligning with the "locally finite partial order" structure in causal set theory. Furthermore, we define the "Universe QCA Object" \mathfrak U_{\rm QCA}=(\Lambda,\mathcal H_{\rm cell},\mathcal A,\alpha,\omega_0), provide an equivalence theorem of "QCA Locality \Longleftrightarrow Local Finiteness of Causal Partial Order", and construct a 1D Dirac-type QCA in the single-particle limit, demonstrating how to recover the continuous Dirac equation in appropriate scaling limits. Finally, we discuss the expression of observation, entropy, and the arrow of time within this framework, as well as relationships with causal set quantum gravity and discretization schemes for quantum field theory.