Algebraic Theory of Inductive Mathematical Modeling: From Experimental Observations to State-Space Representations

In empirical sciences, the inductive construction of mathematical models relies on observations from controlled experiments, yet the conditions under which such constructions yield well-defined deterministic dynamics remain largely implicit. Within the framework of Axiomatic System Dynamics (ASD), this paper develops an algebraic theory of inductive mathematical modeling based on an ontological–epistemological dual structure. The modeling domain (MD) is formalized as a mathematical structure encoding initial conditions, loading protocols, and admissible loading ranges, thereby characterizing model applicability and the mechanisms of failure under improper extrapolation. Weak and strong validity conditions characterize, respectively, the uniqueness and dynamical closure of observable parameter evolution, yielding an algebraic constraint structure for mathematical models and their induced state-space representations. Generalized state parameters incorporate loading-history information, and an existence theorem is established. Difference representations of states and parameters provide an algebraic basis for expressing the underlying logic of comparative experiments, controlled-variable methods and causal inference. Together, these constructions formalize otherwise implicit concepts and structures in conventional modeling practice, render them amenable to algebraic reasoning, and provide a unified theoretical foundation for future extensions.