Towards a General Theory of Sequence Operators: An Axiomatic Foundation for the Algebraic Dynamics of Generated Sequences

This work establishes a rigorous, axiomatic foundation for a unified theory of sequence operators. The central innovation is a paradigm shift from viewing sequences as static, indexed collections to treating them as dynamic operator states that evolve under branching, contraction, and affine transformation. At the heart of this framework lies the formal definition of a sequence operator system as a septuple S= (G, X, B, W, C, F, ), where G is a generator space, B a sequence-building law, W an abstract algebraic inheritance space equipped with a monoid law and a partial order, C a family of classification schemes that determine the attenuation of inherited properties, F a family of basic operations on generators, and a derived evolution mechanism built from F and C by finite composition and union. The inheritance space is treated as an independent algebraic object whose decay law is entirely controlled by the chosen classification scheme, not imposed as a fixed physical density. We develop the core structural theory of such systems, including a systematic methodology for extracting invariants by analyzing the action of basic operations, a theory of potential functions and compression estimates, and a precise analysis of reachability, fixed points, and branching dynamics. The power and universality of the formalism are demonstrated by instantiating it into canonical model families---arithmetic, geometric, and linear recurrence progressions---recovering their classical dynamics and revealing new structural constraints within a single, coherent algebraic language. This monograph is intended as both a foundational text and a long-term research program, carefully separating proved propositions from formal theorem targets and open problems, to establish sequence operator theory as a new, independent field of mathematical inquiry.