Preparation-Return Holonomy and Equilibrium Uniqueness in Engineered Interacting Networks

Returning a quantum wavefunction and its Hamiltonian to their initial values need not return the configurations guided by that wavefunction. This paper uses the configuration changes left by exact quantum return cycles to characterize the Born probability law at a single preparation. For any prescribed finite connected particle graph, the manuscript constructs an engineered harmonic network with fixed, nonzero, directionally restricted pair interactions. Local one-body scalar controls implement exact nonlinear and Gaussian return protocols. The Born measure is proved to be the unique probability law invariant under the resulting library of configuration maps. The uniqueness theorem covers all Borel probabilities, including singular laws, without assuming a probability density or a regular assignment across wavefunctions. The paper supplies the guided-flow constructions and exact endpoint corrections, establishes a broader Gaussian holonomy result for connected quadratic networks, and develops consequences for reversible preparation transport and calibrated readouts. A randomized return controller is also analyzed. Absolutely continuous initial laws converge to equilibrium in total variation, but a quantitative obstruction prevents exact equilibration after finitely many rounds. Retained command history permits an inverse echo that recovers the initial nonequilibrium. Applying the uniqueness theorem to an actual preparation requires the explicit statistical premise of invariance under the specified returns. The manuscript presents self-contained mathematical results within its stated engineered model; it does not derive that premise or establish laboratory realization.