Slow-Mode Adaptive Memory Gate

Add this mechanism to a recurrent or SSM layer that already maintains a state z_t. At each chunk boundary, rather than recomputing an eigendecomposition, estimate a conservative spectral scale s_t for M using 5 to 10 power iterations; for a fixed learned M this can be cached and updated only once every 128 tokens or once per optimizer step. Set tau_t=-1/log(max(s_t,1e-4)) and detach it from backpropagation. Compute a retention scalar r_t=sigmoid(w0+w1 log(1+tau_t)+w2 delta_t), where delta_t=||q_t-q_{t-1}||_2 normalized by a running RMS. Update the memory as z_new=r_t(C q_t+D z_t)+(1-r_t)z_t. For streaming inference, skip the full memory update when r_t<0.05 and refresh or reset the state when delta_t exceeds a learned threshold; for training, still compute the dense update so gradients remain comparable, or use straight-through skipping. Enforce D stability with D=rho tanh(Draw)/sigma_max(Draw), rho<0.99, and add a penalty if the estimated full-block scale s_t exceeds 0.99. The exact paper-derived computation is the state transition and its exponential decay interpretation; the threshold, retention weights, and spectral estimate are learned or empirical. First evaluate on WikiText-103 or PG19 using a 2-layer 512-width recurrent or SSM language model, against the same model with always-on state updates. Report perplexity at context lengths 512, 2048, and 8192, tokens per second, fraction of skipped updates, and hidden-state norm. A successful result should preserve or improve long-context perplexity while reducing recurrent update FLOPs and eliminating instability caused by near-unit or super-unit modes.