Force--Torque Reciprocity and the Inference of Concealed Dissipation in a Geared Brownian Machine

We show that in a reciprocal Brownian motor the entropy production hidden behind a mechanically stalled coordinate can be reconstructed exactly from measurements of that coordinate alone. We introduce a minimal, analytically solvable Langevin motor in which an observed translational coordinate is coupled reciprocally to a hidden internal rotor: a single periodic potential $V(x-\mathrm{\ell }\theta )$ generates both the force on the observed coordinate and the reaction torque on the hidden one, so that ${\tau }_{\mathrm{i}\mathrm{n}\mathrm{t}}=-\mathrm{\ell }{F}_x$ holds identically. Force–torque reciprocity together with translational symmetry produces a local current identity that closes the hidden thermodynamic bookkeeping. From it we prove that the Harada–Sasa heat measured through the observed coordinate equals the positive current-square dissipation of that coordinate, with the information-flow correction vanishing identically. eciprocal class. ries in that it requires no separation of time scales.