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

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…