Chemical Systems as Ternary $Γ$-Semirings:Theory, Case Studies, and Operational Tests

Chemical systems are traditionally described by lists of species, reactions, and externally imposed kinetic laws, a framework that lacks an intrinsic algebraic structure governing how transformations compose. We propose an axiomatic reformulation in which a chemical system is modelled as a ternary Gamma semiring (TGS), where chemical states form an additive semigroup, mediators encode catalytic or environmental context, and mediated transformations are represented by a ternary operation. We show that the TGS axioms admit direct physical interpretations: distributivity corresponds to ideal, non-interfering parallel reactions, while associativity characterizes thermodynamic path-independence. Classical systems including Michaelis-Menten kinetics, global equilibrium, and allosteric regulation are recovered as different algebraic regimes, and we develop operational tests that quantify departures from the axioms through experimentally measurable indices. The resulting framework unifies equilibrium, kinetics, regulation, and chemical computation within a single algebraic language, offering new principles for the analysis and design of responsive or self-regulating materials.