Dynamically Regulated Topological Transport in a χ-Coupled Floquet Lattice

We introduce a dynamically regulated χ-coupled Floquet-DNLS lattice in which a second-order stiffness field χ simultaneously modulates local dispersion and the instantaneous Berry curvature. A variational control functional penalizing large curvature, non-adiabatic transitions, and gap fluctuations drives χ, yielding a formally coercive energy structure, strict positivity of χ, and energetic protection against gap collapse. Using a semiclassical anomalous-velocity derivation and linear-response theory, we prove perturbative geometric transport enhancement (Proposition 5.1) under adiabatic conditions. Extensive simulations with two independent integrators, ensemble statistics, and a symmetric-drive control experiment confirm a ~4.6× fidelity improvement and ~2.1× larger geometric drift of purely topological origin. A concrete implementation roadmap for AlGaAs waveguide arrays is provided. This framework establishes a new class of adaptive nonlinear topological lattices in which dispersion is dynamically self-regulated while geometric transport is geometrically stabilized.