Topological Interference in Multi-Source Systems and a Topologically Coupled Field Theory

When multiple physical sources (e.g., vortices, magnetic flux tubes, disclination lines) exhibit non-trivial topological correlations (such as entanglement or linking), the fields they excite no longer obey the linear superposition principle. This paper proposes a "Topologically Coupled Field Theory" (TCFT). Starting from first principles, we: (1) define the order parameter field and topological charges; (2) quantitatively characterize the topological correlations among multiple sources using the Gauss linking number; (3) construct an energy functional containing a topological coupling term based on anisotropic effective medium theory, deriving a modified field equation via the variational principle; (4) explicitly solve the equation under the weak-coupling approximation, proving that the total potential field can be decomposed into a classical linear superposition term plus an interference term proportional to the linking number, which exhibits an oscillatory fringe structure in space; (5) explicitly define the theory as a "passive topological approximation" where the topological flow is treated as a rigid background, and delineate its applicability boundaries. This theory provides three falsifiable predictions: a quantitative formula for the deviation of the dual-source potential field from linear superposition, the conservation of total topological charge, and the quantization of the phase field path integral. Experimental verification schemes cover desktop fluid experiments, liquid crystal defect manipulation, and ultracold atomic gases. This work provides a rigorous, finite, and experimentally decidable mathematical framework for understanding and utilizing topological interference phenomena in multi-source systems.