Residual-Update Halting

Use this as an inference-time controller for the bounded increment loop, with an optional compute penalty. Run the recurrent update normally, but after computing h_next save delta=h_next-h and q=norm(delta)/(norm(h)+eps). Maintain a counter stable_count; increment it when q<=tau and reset it otherwise. Once stable_count reaches m, estimate the local spectral margin every few iterations rather than every iteration. Obtain the estimate with one or two Arnoldi steps based on Jacobian-vector products through the recurrent update; for a cheaper first version, use the norm ratio ||Jv||/||v|| as a conservative proxy but report it separately because it estimates operator amplification rather than spectral radius. Halt when both conditions hold, or force termination at T_max. Use tau in {1e-2, 3e-3, 1e-3}, m=2 or 3, r=0.98, and eps=1e-6. During training, sample a random stopping time or use the maximum loop count so all examples receive gradients; at inference, optionally add a compute penalty lambda_T*T_used. Compare against ACT based on a learned halting head and against fixed T on delayed-copy, reasoning, and small language-model validation sets. Measure average loop count, accuracy or perplexity at equal compute, premature-halt rate, and correlation between the stopping decision and the remaining-state error ||h_T-h_{T_max}||. Success is at least 25% fewer loop evaluations with no more than 0.5% accuracy loss and substantially fewer failures caused by activation-norm collapse.