Bridging the Retarded Field and the Finite-Size Soliton in Global Realism: A Unified Closed Expression for the Historical Disturbance Field

The first paper of the global-realism program derived a retarded historical disturbance field from the linearized Einstein equation under a point-particle source approximation. The second paper replaced that point-particle ontology by a finite-size Hopf-soliton ontology for elementary fermions. The present paper performs the explicit mathematical bridge between those two foundational works: it substitutes the normalized finite density profile of the Hopf soliton directly into the retarded Green-function solution already established in the first paper, obtaining a unified closed expression for the historical disturbance field generated by matter particles. The resulting formula applies uniformly to all fermionic sectors whose particle identity is encoded by topological and internal labels. It removes the short-distance singularity inherent in the point-source approximation, preserves the correct far-field asymptotics, and inherits the causal retardation that encodes microscopic memory. The paper further proves that this historical disturbance field itself requires no additional closure assumptions—it is an exact derived object once the source is specified—thereby clarifying the logical status of the later coarse-grained memory-field equations as effective theories obtained under well-defined statistical hypotheses. The bridge formula is presented as a self-contained theorem, with its domain of validity, failure modes, and systematic extension hierarchy explicitly recorded.