Entropy–Topology Unified Framework: A Complex Operator Z = M + iQ for QCD and the Standard Model

We introduce a complex operator Z = M + iQ that unifies dynamical (mass-related or coupling) and topological (charge- or θ-related) contributions into a single holomorphic structure. This compact formulation offers profound advantages compared to separate treatments of M and Q. It naturally incorporates U(1)-phase symmetries, simplifies the construction of invariants, and allows for the use of complex-analytic methods. When applied to the Standard Model (SM) and quantum chromodynamics (QCD), the complex operator provides a unifying perspective on renormalization group flows, confinement, CP violation, and the grand unification of couplings. Within a diagram-Hilbert space interpretation, the convergence of SM couplings appears as a holomorphic projection, linking topological and dynamical aspects. This framework extends existing approaches to unification by combining running couplings and θ-angles in a single holomorphic fixpoint condition.