Control Consistency and Born Equilibrium: Uniqueness from Local Potentials and Fixed Interactions

What singles out the Born probability distribution in Bohmian mechanics when the same probability assignment must remain consistent across different physical controls? This paper proves a conditional uniqueness theorem for quantum equilibrium in nonrelativistic interacting systems. Under explicit regularity assumptions, a normalized probability assignment depending only on the current wavefunction must equal the Born distribution if it is consistent with Schrödinger evolution and Bohmian guidance across the admitted local scalar potentials. The pair interactions remain fixed, connect the particles, and have genuine mixed spatial dependence. The proof does not require additional controllable many-body potentials. The required controls can be reduced to translates and real amplitudes of a single suitable spatial profile for each particle. The manuscript also provides extensions to effective spin and identical-particle sectors, a separate characterization based on binary-flag heredity, and continuity conditions extending the results to wavefunctions with nodes. Consequences for apparatus records are stated under explicit preparation and measurement-coupling assumptions. The result characterizes the Born law within the stated statistical framework. It does not derive the control-consistency premise from deterministic dynamics or establish relaxation from arbitrary initial distributions. This revised preprint presents the theorem statements and proofs in a self-contained form for independent scholarly assessment.