Projected Play-State Memory

Insert the module as a recurrent memory update in an SSM, GRU-like block, or recurrent policy. Let the network produce an unconstrained drive increment Delta_pi_t=g_theta(x_t,h_{t-1}), and maintain a latent state sigma_t in a simple convex set C, initially a box |sigma_i|<=b_i or an l2 ball. Compute u_t=sigma_{t-1}+Delta_pi_t, then project with sigma_t=P_C(u_t). For a box, projection is elementwise clipping; for an l2 ball, use P_C(u)=u min(1,b/||u||_2). Feed sigma_t to the output head and optionally expose the correction r_t=u_t-sigma_t as a second channel, analogous to an elastic/plastic decomposition. Use a learned positive diagonal metric by scaling coordinates before projection, or begin with the Euclidean metric to avoid instability. The paper supplies the projection, normal-cone characterization, and nonexpansive guarantee; the set C, metric, drive network, and whether to expose r_t are engineering choices. Compare against the same backbone with an unconstrained additive state update and identical parameter count and truncated-BPTT length. Start with copy, adding-long-memory, and sequential-MNIST tasks using a two-layer GRU or SSM. Track hidden-state norms, gradient norms, input-perturbation amplification, validation accuracy, and performance when test sequences are much longer than training sequences. Success means bounded states and lower perturbation amplification without loss of short-sequence accuracy, together with improved length extrapolation.