Multiplicative Manifold Unscented Recurrent Cell

1. Integration point: use this as a replacement for the hidden update in a GRU-like recurrent model, a learned Kalman filter, or a latent state-space model. Reserve three latent coordinates for a quaternion and keep the remaining d coordinates Euclidean. Let a small MLP produce a deterministic transition and process covariance from the current latent state and input: (x_bar, log_Q) = F_phi(x_t, u_t). 2. Pseudocode: maintain q_hat, z_hat, and P. Compute L = Cholesky((n + lambda)P), create the zero sigma point and the positive and negative columns of L, and split every tangent point into delta_theta and delta_z. Retract the rotational part using delta_q(delta_theta) multiplied by q_hat, add delta_z to z_hat, and run the same neural transition on every sigma point. Estimate the quaternion mean by selecting an initial sigma-point quaternion and iteratively averaging weighted logarithmic rotation errors relative to that mean. Compute Euclidean means directly, express every propagated quaternion as a tangent residual relative to the mean quaternion, form P_plus with the displayed covariance equation, and add a softplus-parameterized Q. Normalize quaternions after neural propagation. Train with next-step negative log likelihood, supervised state loss, or both. 3. Computed quantities are sigma points, Cholesky factors, quaternion retractions, tangent logarithms, and covariance updates. The parameters alpha, beta, and kappa are chosen hyperparameters; Q can initially be a learned diagonal matrix and later be calibrated from residuals. If backpropagation through Cholesky is unstable, stop gradients through P for the first experiment. 4. First cheap experiment: create synthetic rigid-body sequences with known quaternion rotations plus a 32-dimensional Euclidean latent signal. Compare an ordinary GRU, an additive UKF-GRU, and this multiplicative UKF-GRU at transition intervals Delta_t in {1, 2, 4, 8, 16}. Measure quaternion geodesic error, covariance NEES = e_t^T P_t^(-1) e_t, the fraction of invalid quaternions, and the fraction of divergent runs. The quantitative prediction is that the multiplicative model preserves unit-quaternion validity exactly and maintains NEES near the tangent dimension n at larger Delta_t. The additive or first-order baseline should show a sharp consistency failure when the one-sigma rotational spread sqrt(trace(P_theta_theta)) becomes approximately 0.3 radians or larger; the measured divergence boundary should shift with this curvature scale.