We derive a non-equilibrium efficiency bound for processes that displace excitations against an effective gravitational potential in a Bose–Einstein condensate (BEC) analog-gravity system. Applying the Clausius inequality to a two-temperature process—a system at the local analog Unruh temperature T coupled to a reservoir at Tres, with the screen entropy change taken from Verlinde's expression—under the explicit assumption that the screen exchanges heat locally reversibly (δQ = T dS) while the reservoir coupling carries the irreversibility, gives the Carnot bound η ≤ 1 − T/Tres. For laboratory BEC parameters (Tres ~ 50 nK, Tanalog ~ 10 nK) this predicts η ≤ 0.80, a 20% suppression below the naive η ≈ 1. The regime T/Tres ~ O(1)—accessible in BEC analog gravity but not in any astrophysical setting—is where the bound has empirical content. The central substitution δQ = T dS is examined through Lindblad master-equation simulations across five settings; for thermal-like initial states it is an algebraic identity, so these runs are consistency checks on the integrator rather than independent tests, and the one coherent-initial-state setting that leaves the thermal manifold makes it fail, delimiting the scope. We do not modify gravity or claim entropy–gravity unification; we claim only a falsifiable cold-atom signature distinct from naive energy accounting.
Grant Lavell Whitmer III (Tue,) studied this question.