Functional Agreement of Computationally Equivalent States

Learned stateful sequential models can implement the same semantic computation through different histories but arrive at different internal states. The behavior of these states may diverge during subsequent model rollouts. We introduce Functional Agreement of Computationally Equivalent States (FACES), an auxiliary loss that compares their future behavior under the same continuation. This objective encourages future behavioral agreement without forcing the internal states themselves to be equal. We introduce several FACES variants that target agreement at different future depths. Geometrically, we show that the objective induces a local quadratic form that weights state-perturbation directions according to their effects on future behavioral readouts. We first perform a mechanistic study on simple affine programs and show that learned actions can support much more accurate long-horizon execution under exact affine dynamics. We also show that route-induced state differences can have strongly direction-dependent effects on future behavior. We then show that FACES reduces long-horizon prediction error relative to ordinary supervised training and immediate agreement on controlled visual $\mathrm{SE}(2)$ transformation sequences. Using real-world data, we show that multiscale AFACES achieves the lowest mean test error among the evaluated methods across temporal factorizations of the EuRoC IMU trajectories. We also show that continuation agreement improves over immediate agreement on a discrete, noninvertible problem. These results show that continuation-based behavioral regularization can improve both the interchangeability of states representing the same semantic computation and task performance.