Invariant-Sphere Recurrent State

1. Integration point: use this as the hidden transition of an RNN or state-space layer. Partition a hidden vector into a cyclic ring, compute $Q_i=ah_i^3+bh_i^2h_{i+1}+ch_i h_{i+1}^2+dh_{i+1}^3$, and add input injection separately. Set $\lambda>0$ and initialize coefficients so the sampled radial form is negative. Use a sufficiently small solver step, or compare against an explicit projection to radius $r_*=\sqrt{\lambda/\kappa}$. 2. Pseudocode: u=h/(norm(h)+eps); Q=ring_cubic(h); hdot=lambda*h+Q+W_in*x_t; h=h+dt*hdot; loss=task_loss. Add radial_penalty=(mean(inner)+kappa)^2+variance(inner), where inner=dot(u,Q(u)) over random unit-vector probes. Optionally parameterize $c$ around the predicted boundary $c=-1$. 3. Compute directly from the mechanism the cubic field, Lyapunov derivative, predicted radius, and pole eigenvalue. Estimate empirically the radial coefficient $\kappa=-\operatorname{mean}_{u\sim S^{n-1}}\langle u,Q(u)\rangle$, its variance, the recurrent Jacobian spectral radius, and norm relaxation rate. 4. First experiment: compare equal-width vanilla tanh RNN, orthogonal RNN, and this invariant-sphere RNN on copy-memory and adding tasks at sequence lengths 100, 500, and 1000, using equal parameter counts and optimizer budgets. Include ablations removing the radial penalty and replacing cubic ring coupling by a dense transition. 5. Pre-register the signature: with zero input, the hidden norm should converge to $r_*$ and satisfy $|r_t-r_*|\approx C\exp(-2\lambda t)$ in continuous-time units. Sweeping $c$ should change the pole-direction multiplier at $c=-1$. For Euler integration, the radial linearization predicts a step-size boundary near $dt_c\approx 1/\lambda$. Reject the transfer if the radial form is positive on more than 1 percent of sampled sphere directions, if the measured relaxation rate differs from $2\lambda$ by more than 30 percent, or if norm variance is not reduced by at least 50 percent without improving long-horizon accuracy.