Holonomy-designed recurrent memory

Integrate this into a small GRU-like recurrent block or an SSM state update, not into the output head. Split the hidden state into two low-dimensional factors: d_t for normalization/gain and w_t for winner identity. Compute continuous updates h'_t = RNN(h_{t-1},x_t), then map them to soft discrete states p_t = softmax(A h'_t / tau), with p_t containing logits over the finite joint states q=(d,w). Implement each input bin u as a transition matrix T_u over joint states, or parameterize T_u through differentiable DN and WTA updates; use p_t = p_{t-1} T_u during the algebraic training branch. Pseudocode: initialize p0; for each symbol u, p <- p @ T[u]; retain the resulting p for the cycle loss; decode h_t from p or use the continuous hidden state as the task representation. Select a target pair q0,q1 and two short input words a,b, then add the displayed cycle loss so T_b T_a maps q0 to q1 and the reverse word maps q1 to q0. Also enumerate all states and generators after training, construct M by breadth-first closure under composition, and reject candidates with unwanted long cycles or excessive nonabsorbing states. The paper's exact algebraic contribution is the finite transformation-monoid closure and the Z2 restriction test; transition probabilities, temperature, and task loss are engineering choices. Estimate robustness by perturbing logits and measuring whether the decoded state remains in C. First experiment: train a 2-layer vanilla GRU and this 16- or 32-state algebraic cell on a delayed-copy task with distractor symbols. Compare accuracy, hidden-state corruption under noise, and number of training examples needed. Success means higher delayed-copy accuracy and a visibly stable two-state orbit, without increasing parameter count by more than 2x.