Operational Regimes of Ternary $\Gamma$-Semiring Chemical Systems

Abstract

Ternary $\Gamma$-semiring chemical systems provide an algebraic language in which mediators such as catalysts, solvents, inhibitors, thermodynamic conditions, and external fields enter the transformation law as internal arguments. Building on the published MATCH foundation article, this paper studies operational regimes of the compressed TGS notation rather than proposing a second axiomatization. Distributivity is interpreted as observable non-interference of mediated channels, while associativity is interpreted as observable path-independence of bracketed mediated protocols. Since the state space is only semigroup- or semiring-level, subtraction is not used at the state level; instead, observable defect and ratio indices are defined after applying a non-negative observable. We prove that distributivity implies observable non-interference under observable additivity, and that associativity implies observable path-independence. Nonzero defects detect observable distinguishability. To strengthen the diagnostic layer, we prove recovery theorems showing that separating families of observables can recover algebraic distributivity and associativity from vanishing observable defects. We also introduce approximate operational regimes and prove ratio bounds controlled by uniform defect estimates. Under strict convexity, uniqueness, and common global-constraint assumptions, a potential-generated equilibrium operation is shown to be associative. A symbolic intermediate-dependent Michaelis--Menten type construction shows how non-associativity may arise in kinetic regimes, and regulatory/allosteric-type non-distributivity is discussed cautiously through observable coupling.