Turing Machines and Cellular Automata via Zeta Functions: From Computational Ontology to Holographic Construction

Within the Zeta holographic framework, this paper systematically elucidates the deep mathematical relationship between Turing machines and the Riemann zeta function, demonstrating how to construct cellular automata through zeta functions. Core discoveries include: (1) Through Voronin's universality theorem, all computable functions of Turing machines can be encoded by the zeta function within the critical strip; (2) Precise correspondence exists between Turing machine state transition matrices and Euler product representations; (3) The finite state nature of Turing machines corresponds to projections of the infinite-dimensional spectrum of zeta functions in Hilbert space; (4) Based on zeta function zero distribution, cellular automata with quantum properties can be constructed; (5) Turing's historical work in 1950 verifying the Riemann Hypothesis using the Manchester Mark 1 computer reveals the essential unity of computation and number theory. This paper establishes a complete mathematical bridge from Turing computability to zeta function analytic properties. It proves that Turing machines are finite-dimensional substructures of zeta functions. This unified framework not only resolves the deep connection between the halting problem and the Riemann Hypothesis, but also provides a novel number-theoretic foundation for quantum computing and cellular automata theory. Through the information conservation law $\cI_{\text{total}} = \cI_+ + \cI_- + \cI_0 = 1$, we reveal the precise balance mechanism between positive information generation and negative information compensation in computational processes, providing a unified computational ontological interpretation for understanding the Casimir effect, quantum fluctuations, and path integrals in general relativity.