A DVR-Based Framework for Detecting Wave–Particle Duality Transitions and the Quantum Cheshire Cat Effect

We introduce a novel variational framework for identifying the boundary between quantum interference and classical determinism using Difference-based Variational Reconstruction (DVR). By independently analyzing DVR residuals for particle trajectories and their associated magnetic moment (spin), we uncover a computational signature of the quantum Cheshire Cat phenomenon, where a particle and one of its properties appear to localize in different regions. In our framework, energy variance acts as a physical observable to quantify the transition from a broad quantum superposition to a dominant classical path, while DVR residuals quantify the local reconstructability of motion and internal properties. We show that the DVR residual for spin reconstruction peaks when the spin reversal occurs mid-path, while the trajectory remains classically smooth --- revealing a quantitative mismatch in reconstructability between particle and property. Furthermore, we demonstrate that DVR residuals grow under simulated decoherence (via noise), offering a natural diagnostic tool for coherence breakdown. Our method bridges path integral formulations with a variational, coefficient-based representation of physical trajectories, recovering semiclassical paths from within the full quantum superposition. This framework unifies and extends concepts from quantum mechanics, control theory, and plasma dynamics into a single path-based computational model.