O-Regular Mappings on C(C): A Structured Operator–Theoretic Framework

Motivation. Analytic function theory on commutative complex extensions calls for an operator–theoretic calculus that simultaneously sees the algebra-induced coupling among components and supports boundary-to-interior mechanisms. Gap. While Dirac-type frameworks are classical in several complex variables and Clifford analysis, a coherent calculus aligning structural CR systems, a canonical first derivative, and a Cauchy-type boundary identity on the commutative model C(C)≅C4 has not been systematically developed. Purpose and Aims. This paper develops such a calculus for O-regular mappings on C(C) and establishes three pillars of the theory. Main Results. (i) A fully coupled Cauchy–Riemann system characterizing O-regularity; (ii) identification of a canonical first derivative g′(z)=∂x0g(z); and (iii) a Stokes-driven boundary annihilation law ∫∂Ωτg=0 for a canonical 7-form τ. On (pseudo)convex domains, ∂¯-methods yield solvability under natural compatibility and regularity assumptions. Stability (under algebra-preserving maps), Liouville-type, and removability results are also obtained, and function spaces suited to this algebra are outlined. Significance. The results show that a large portion of the classical holomorphic toolkit survives, in algebra-aware form, on C(C).