Conservative Quantum Phase-Field Duality for Elastic Coherence in Mesoscopic Systems

We formulate a conservative, [Formula: see text]-invariant, revisionblueHamiltonian extension of Schrödinger/Gross–Pitaevskii hydrodynamics in the rest frame of an effective coherent medium in which the complex matter field [Formula: see text] couples to a real, dimensionless phase-stiffness field [Formula: see text]. The auxiliary field dynamically modulates the Bohm quantum potential through a coupling proportional to the local Fisher-information density, while itself evolving through an inertial equation driven by amplitude gradients. revisionblueThe variationally consistent amplitude-gradient coefficient is [Formula: see text], so positive excursions of [Formula: see text] produce an adaptive softening of quantum pressure. From a single variational principle we obtain coupled evolution equations and revisionbluea norm-conserving energy functional that is lower bounded in the positive-stiffness sector. Standard Schrödinger/Gross–Pitaevskii dynamics is recovered when [Formula: see text], whereas the classical Hamilton–Jacobi limit requires both [Formula: see text] and the disappearance of the auxiliary excitation. Linear analysis reveals an independent gapped stiffness mode in addition to the Bogoliubov branch, whose zero-momentum gap and quadratic dispersion coefficient determine the parameter ratios [Formula: see text] and [Formula: see text] governing restoration, smoothing, and inertia. In the nonlinear regime the same feedback revisionblueproduces an oscillatory auxiliary-field response and a numerically converged, parameter-dependent reduction of wave-packet spreading. The packet remains dispersive rather than forming an interaction-free self-bound state. The framework further predicts revisionbluea conditional logarithmic visibility relation under a calibrated interferometric readout and reversible finite-rate response loops under cyclic driving. These loops quantify conservative inertial lag rather than dissipated energy or intrinsic hysteretic memory. These results establish the phase-field duality as a minimal and falsifiable Hamiltonian extension in which coherence behaves as an elastic medium.