Dynamic Topology: Phase Closure Condition as Topological Invariant and First Principle of Connectivity

Abstract

Abstract The core paradigm of classical topology---invariance under continuous deformation---is essentially a "God's-eye view": it requires the observer to stand outside the space, overlook the entire structure at once, which leads to the need to traverse all closed paths to determine the presence of holes, resulting in exponential computational complexity. This paper proposes Dynamic Topology, which elevates the phase closure condition rigorously derived in Dynamic Complex Analysis to the first principle of topology, replacing the "God's-eye view" with the "ant's perspective"---an ant only needs to walk along a closed path and measure whether the phase closes to know the global topology. A hole is a region where the phase cannot close (I ≠ 0), connectivity corresponds to the integrability of phase differences, genus corresponds to the number of independent non-zero topological charges, and the fundamental group and homology group correspond to the algebraic structure of phase winding numbers. Under emergent continuity, these dynamic topological definitions are strictly equivalent to classical static topological definitions. Conservation of topological charge is guaranteed by emergent differentiability---an intrinsic property of generativism, not an extra assumption. Computationally, it offers a revolutionary advantage: all topological properties can be obtained by O(1) computation of local phase gradients, without global traversal. Biological verification provides independent support---phenomena such as the anticipatory behavior of slime molds without neurons and distributed phototaxis in soft robots are instantiations of dynamic topology in living systems. The resulting generative intelligence spectrum (L0--L4) proves that neuronless intelligence and neuronal intelligence share the same first principle. Core new contribution: This paper further establishes a rigorous theorem for the necessary emergence of topological constraints under strong coupling---topological constraints are not the patent of "special systems" but the inevitable signature of Axiom 4 coupling dynamics in the strong-coupling limit. The historical term in the phase coupling equation encodes topological constraints as queryable algebraic structures, revealing the deep origin of O(1) topological computation---the topological charge is not "computed" but "read from historical records". A hole is not an "inherent property" of space but a "survival trace" of the phase field's inability to close. The ant does not need God's-eye view---by walking once around, it knows the global topology. Keywords: dynamic topology; phase closure condition; topological charge; connectivity; ant's perspective; O(1) topology computation; neuronless intelligence; generative intelligence spectrum; strong coupling; inevitability of topological constraints

The authors' abstract, as published at the source. Zenodo (CERN European Organization for Nuclear Research), 2026 · DOI ↗