Abstract
<title>Abstract</title> <p>Topology is essential in vortex dynamics, but a publishable fluid-mechanical theory must state exactly where topology enters, what it predicts, and what would falsify the claim. This paper formulates a bounded projected-response framework in which the Navier–Stokes equations remain the local physical dynamics. Topology enters only after a velocity field or coherent-vortex state is mapped to declared topology-state coordinates, a structural free energy is defined on those coordinates, and the resulting reduced gradient is projected to a measured observable. The central claim is therefore not a universal topological body force, but a response-level statement: topology-state coordinates can carry predictive information lost by gross geometry or total helicity, and coherent vortex branches can expose measurable projected response gaps. Two evidence loops are reported. First, controlled incompressible vortex benchmarks show that topology-enriched coordinates improve declared reduced observables beyond geometry-only and global-helicity baselines, including a final helicity-concentration prediction and an anti-parallel reconnection-window stress test with a topology-column permutation control. Second, a public dynamic-stall-vortex branch is used as a case study of one-mode finite-capacity response; an amplitude-canceling three-frequency ratio isolates a characteristic response scale before fitting the saturation amplitude. The combined result supports a limited but falsifiable interpretation: topology-state free-energy gradients are useful as projected response coordinates for coherent vortex morphology. The paper does not claim a replacement for Navier–Stokes dynamics, a full- field DNS closure, a universal aerodynamic constant, or a direct measurement of a microscopic topological force density.</p>