This monograph establishes a rigorous axiomatic foundation for a theory of multivariate sequence operators. The central innovation is a formalism in which several distinct classification mechanisms act simultaneously on a common generator space, each governed by its own inheritance structure and attenuation law, yet globally coupled through the shared generator. The primitive object of study is introduced as a coupled unary operator family of dimension n, consisting of n independent unary coordinate systems that share the same generator space, value map, and basic operation family. Each coordinate carries its own inheritance space Wⱼ, classification scheme Cⱼ, and unary evolution map ⱼ. The multivariate weighted state space is defined as the Cartesian product A = G ₉=₁^n Wⱼ, and the one-step coupled evolution operator: A P₅₈₍ (A) is built by serialized single-coordinate activation: at each step, exactly one coordinate is activated, its unary rule is applied, its weight is attenuated, and all other weights are held fixed. The resulting quantum coupling tree T (₀) is a rooted, locally finite, edge-labeled tree whose vertices are admissible histories of coordinate-branch choices and whose state labels evolve by the one-step coupled rule. This tree is shown to be noncommutative in general, with the source of noncommutativity rooted in the fact that the first activation changes the common generator on which the second activation acts, so that the global coupled system is irreducible to a Cartesian product of independent unary trees. The framework also establishes an exact projection theory: when a descendant subtree is closed on a subset J of coordinates, the complementary coordinates freeze as constant parameters, and the active dynamics on J reduce canonically to the corresponding lower-dimensional coupled system. A strict morphism theory for parallel coupled families is developed, including embeddings, quotient morphisms, regular strict quotient morphisms, and isomorphisms, together with transfer theorems for invariants, compression potentials, and reachable realized-state graphs. Model families—arithmetic, geometric, and linear-recurrence—are instantiated in full detail, demonstrating the operational validity of the axioms and yielding precise commutativity and noncommutativity criteria. Throughout, a step-by-step writing blueprint and a unified logical-status labeling system keep the axiomatic foundations, structural development, concrete models, and long-term research targets sharply separated. The monograph is intended as both a foundational text and a long-term research program for the study of coupled evolutionary structures in discrete dynamical systems.
Jianming Wang (Sun,) studied this question.