Abstract and publication identity

An exact return of a quantum state need not return the configurations guided by its intervening wavefunction. We use this distinction to characterize quantum equilibrium at a single preparation. For any prescribed finite connected particle graph, we construct a positive harmonic holding Hamiltonian with fixed nonzero, directionally restricted pair interactions. Two coordinates of one particle form a reserved nonlinear plane. Physical one-body scalar controls implement exact state-returning density loops in that plane and exact Gaussian returns whose configuration maps realize $SO(3N)$ in whitened coordinates. The only Borel probability invariant under the resulting return library is the Born measure. No density, absolute continuity, or regular probability functional across wavefunctions is assumed. The proof supplies global guided flows, a continuum inverse-engineering construction, and finite-dimensional endpoint correction preserving the Gaussian configuration curvature. We also prove a broader Gaussian holonomy theorem for connected fixed quadratic networks, without extending the nonlinear conclusion to that larger class. Reversible preparation paths and fixed-network linear readouts give corresponding transport and no-hysteresis statements. Applying uniqueness to an actual preparation law requires the separate statistical premise of invariance under the specified returns. We also analyze a lazy inverse-balanced randomized return controller: its absolutely continuous laws converge in total variation, but every finite round obeys a reverse bound, and retained command history can recover the initial nonequilibrium by an inverse echo. These controller results do not derive the required statistical premise.