Holonomy-Attractor Recurrent Cell

Use this mechanism in a 2D or 8D latent RNN or state-space model, initially with a 2D diagnostic subspace. Parameterize the recurrent transition in homogeneous coordinates. Given hidden state h, input x, and unconstrained scalars s and t produced by a small input network, construct M(s,t), apply its affine action to h, add a bounded input residual, and use a sigmoid gate to interpolate with the previous hidden state. For higher-dimensional states, apply the same cell to disjoint two-dimensional planes or to a learned two-dimensional projection P h, while retaining an orthogonal residual branch for expressivity. Pseudocode is: compute s and t; build M(s,t); compute the affine update h_bar; set h_new equal to (1 minus gate) times h plus gate times tanh(h_bar); accumulate products of the linear parts and translations using homogeneous matrix multiplication; compute a Jacobian-vector product J_k u; and add the projective loss. The group matrix and two-basin limit are taken from the paper, while X, omega, and finite-time convergence rates are estimated from trajectories. Start with sequential MNIST or a length-512 copy task using a 2D vanilla tanh RNN, the structured cell, and the structured cell without the projective loss. Perturb hidden states and measure angular error to X versus cumulative q_k=omega(T_{1:k}). The falsifiable prediction is monotone angular alignment in each sign basin as q_k grows, with failure when q_k changes sign or remains bounded. Also measure long-horizon perturbation variance and compare its decay against an exponential or power-law fit before evaluating task accuracy.