Finite-Horizon Control Consistency for Rank-Deficient Diffusion Bridges

Control-consistency methods for diffusion bridges recover state-space covectors from Brownian-space controls using instantaneous diffusion geometry, which becomes singular under rank-deficient noise even when the drift propagates forced directions into unforced coordinates. We introduce Finite-Horizon Bridge Control (FBC), replacing singular pointwise recovery under rank deficiency with a finite-horizon controllability lift built from dynamically transported noise directions. We show that the self-consistency identity itself does not require full-rank diffusion and connect the finite-horizon Gramian to Malliavin covariance. We prove exact finite-horizon recovery for controllable time-inhomogeneous affine diffusions with deterministic coefficients and characterize the information scales of dynamically reachable directions over short horizons in the local linearized controllability geometry. For nonlinear systems, we give sufficient executable-consistency conditions and an explicit short-window convergence rate for underdamped Langevin dynamics. FBC thus extends control consistency to rank-deficient and hypoelliptic diffusion bridges by exploiting finite-time controllability.