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The proposed theory of causally structured discrete fields studies integer values on directed edges of a self-similar graph with a propagation rule, which we define as a set of valid combinations of integer values and edge directions around any vertex of the graph. There is an infinite countable number of variants of the theory for a given self-similar graph depending on the choice of propagation rules, some of these models can generate infinite uncountable sets of patterns. This theory takes minimum assumptions of causality, discreteness, locality, and determinism, as well as fundamental symmetries of isotropy, CPT invariance, and charge conservation. It combines the elements of cellular automata, causal sets, loop quantum gravity, and causal dynamical triangulations to become an excellent candidate to describe quantum gravity at the Planck scale. In addition to the self-consistent generation of spacetime and metrics to describe gravity and an expanding closed Universe, the theory allows for the many-worlds interpretation of quantum mechanics. We also demonstrate how to get to unitary evolution in Hilbert space for a stationary Universe with deterministic propagation.