Phase-Polytope Robust Neural Dynamics

1. Integration point: maintain M lightweight phase heads or M copies of a small state-space transition. Heads may share hidden features. Each head is trained on data aligned to one cyclic offset. During training, compute all heads on the same minibatch and form both the centroid prediction and an adversarial convex mixture. 2. Pseudocode: compute predictions F[1:M]; set F_bar=mean(F); compute vertex losses l[m]=loss(F[m],Y); choose m_star=argmax_m l[m], or perform 3-5 projected-gradient ascent steps on alpha over the simplex; set F_mix=sum_m alpha[m]*F[m]; backpropagate loss(F_bar,Y)+lambda_rob*loss(F_mix,Y). At inference, report F_bar and the empirical spread across vertices. 3. Computed versus estimated: the simplex, centroid, and vertex losses are exact; the variance-reduction law is a hypothesis requiring empirical testing. Avoid averaging raw weights unless heads are explicitly aligned; use output-space mixing otherwise. 4. First cheap experiment: use a synthetic M=4 multirate nonlinear oscillator with additive observation noise and compare a single phase head, centroid prediction, ordinary deep ensemble, and centroid-plus-worst-vertex training using the same GRU or neural state-space backbone. Measure rollout RMSE, calibration, and maximum per-phase validation loss. The predicted signature is centroid error variance decreasing approximately as 1/M when phase errors are weakly correlated. Robust training should reduce the maximum vertex loss, and the worst-case loss should approach the average loss as phase discrepancies shrink. Failure of the 1/M trend or persistent large vertex spread falsifies the ensemble interpretation.