Curvature Sign Determines Geometric Error-Propagation Regimes in Behavioral Cloning

Behavioral cloning (BC) optimizes one-step prediction, but deployment feeds predictions back into future states, so local errors compound. We ask how rollout geometry determines the regime in which these errors accumulate without online correction. On homogeneous constant-curvature manifolds, we derive a parameter-free local recurrence separating inherited-error propagation from fresh-error injection. Under direction averaging and zero state-dependent policy amplification ($L=0$), the reduced recurrence yields three regimes: a bounded squared-error floor under positive curvature, diffusive $\sqrt{T}$ root-mean-square growth in flat space, and exponential amplification under negative curvature; a conditional orientation band extends the result beyond exact averaging. Simulations match the theory within 3% in second moment on ${\mathbb{S}}^2$ and ${\mathbb{R}}^2$, and $\mathrm{SO}(3)$ exhibits the predicted stabilization. Learned-policy evaluations on $\mathrm{SO}(3)$ and in an articulated execution chain show that matched one-step error can coexist with different rollout errors, consistent with measured propagation response. At fixed decision/noise-injection counts, longer action chunks strengthen positive-curvature contraction and negative-curvature amplification, while the negative-curvature regime offers a geometric perspective on fragility in hyperbolic reinforcement learning. Curvature sign sets the accumulation regime, so sequential evaluation should measure propagation, not just error magnitude.