Time-Varying Holonomy: The Dynamical Bridge from Local Phase Coupling to Global Structure

Abstract Theorem 2.2 of Topology as a Bridge proves that a locally coupled network necessarily converges to a stable topology—all closed paths eventually satisfy the phase closure condition. However, this conclusion is qualitative: it asserts "inevitable convergence" but does not answer "how"—does the phase accumulation along a single closed path decay monotonically or converge in an oscillatory manner during evolution? Is the gradient flow of local coupling actively "suturing" topological obstructions? Under the axiom system of generativist mathematics, this paper introduces time-varying holonomy Hol_φ(C, T) as a new dynamical object, and strictly distinguishes its two layers: dynamical holonomy (alterable by gradient flow) and topological holonomy (conserved by gradient flow). The core contribution is the proof of the Evolution Direction Theorem: for closed paths with zero topological holonomy, the gradient flow forces the square of the dynamical holonomy to decrease monotonically; for paths with non-zero topological holonomy, the dynamical holonomy tends to the value 2πn corresponding to the topological holonomy. This theorem shows that the gradient flow is actively "suturing" dynamical incompatibility, while respecting the conservation of topological obstructions—the network attains a steady state of "local compatibility, global topological protection" in the limit. Although time-varying holonomy intuitively corresponds to the holonomy of modern differential geometry, it possesses an independent mathematical definition and dynamical content. It presupposes no smooth manifold, no connection form, and no continuous parameterization—it relies only on the phase field on an information pixel network and the local coupling equation. Its core characteristics—discreteness, principal value jumps, gradient flow evolution, topological conservation—have no counterpart in the holonomy framework of modern mathematics. This paper and Analysis Is Topology form a complete complement to the "analysis-topology" sector of the L2 layer: the former establishes the static equivalence between gradient flow steady states and topological closure, while the latter tracks the dynamic evolution trajectory of time-varying holonomy. The results of this paper simultaneously provide a dynamical explanation for the zero distribution of the Network ζ-Operator, a convergence guarantee for the gradient evolution operator of Hexad Computation, and a micro-engine for the methodology of Computation as Proof, with the entire theory relying on L4 Topological Information Theory to close the information-theoretic self-consistency loop—each step of the evolution in the Evolution Direction Theorem is "proving" that incompatibility is being eliminated. Keywords: time-varying holonomy; dynamical holonomy; topological holonomy; Evolution Direction Theorem; local coupling; phase closure; gradient flow; generativist L2 layer; topological information theory