Beyond Trajectories and Jacobians: Distilling Finite-Time Tangent Cocycles for Chaotic Forecasting

State-space losses and local Jacobian objectives do not directly constrain how infinitesimal errors are transported over multiple steps in a chaotic system. We introduce BundleDistill, a training-only objective that distills low-rank finite-time tangent-cocycle geometry from controlled antithetic trajectory rollouts, without requiring governing equations or derivative labels. The student matches cocycle action, probed singular growth, and transported-subspace orientation using Jacobian–vector products and small endpoint factorizations. Under matched optimizer-update budgets across five locked seeds on Lorenz–63, Lorenz–96, and Kuramoto–Sivashinsky, BundleDistill has the smallest observed transported-subspace angle on all three systems. Against the strongest scalar-growth control, restricted-mean valid prediction time increases on Lorenz–96 by 0.018 Lyapunov times (95% percentile interval [0.016,0.021]) and by 0.180 on Kuramoto–Sivashinsky ([0.149,0.215]), while the Lorenz–63 interval spans zero. Five-seed ablations on Lorenz–96 and Kuramoto–Sivashinsky show that component effects depend on the system and endpoint. The auxiliary objective leaves model size unchanged and adds no inference operation, but it increases training cost; under a calibrated optimizer-update-time control, the faster Rand-Jac2 baseline is stronger on Lorenz–96 VPT and geometric endpoints. Formal out-of-regime tests yield no Holm-corrected rejection, and held-out long-run statistics vary by system. The evidence supports finite-time tangent transport as a useful, compute-dependent training target, with its clearest gains on the two higher-dimensional systems studied.