Integrable Elliptic Memory Cell

1. Integration point: use this transition as a recurrent or state-space layer before a task-specific MLP or linear readout. Partition a hidden vector of size 4m into 2m planar pairs, with one learnable angle per pair. Parameterize each angle as theta_j=pi*tanh(a_j), or generate it from a slowly varying context embedding; never form an unconstrained recurrent matrix. 2. Pseudocode: initialize h=Linear(x_0). For each input x_t, compute an input injection b_t=B x_t, set c_j=cos(theta_j) and s_j=sin(theta_j), update each pair by p'_j=c_j p_j-s_j q_j and q'_j=s_j p_j+c_j q_j, then set h_{t+1}=R h_t+b_t and y_t=C h_t. 3. Computed exactly: rotations, their orthogonality, and the predicted invariant. Estimated empirically: gradient norms, memory retention, and the influence of input injection. Monitor DeltaV_t=||h_{t+1}||^2-||h_t||^2 separately with and without b_t. 4. First cheap experiment: compare this cell with vanilla RNN and GRU at hidden width 128 on sequential MNIST and the copy-memory task at sequence lengths 1,000 and 10,000. Use identical Adam settings and input projections. The predicted signature is |DeltaV_t|/V_t below 10^-5 for the autonomous recurrence over 10,000 steps, while unconstrained RNNs show exponential growth or decay when their recurrent spectral radius differs from one. For sinusoidal input, amplitude drift should remain below 1% over 10,000 autonomous steps.