Wasserstein-Gated Cached Equilibrium Adapter

Integrate the module between the encoder or SSM and the prediction head of a small sequence model. Let h_t be the encoded recent history, x_t the current latent state, and y_t a d-dimensional response vector used by the head or policy. Maintain a cache of tuples (z_i,x_i,y_i), where z_i is a normalized history embedding and y_i is the fully recomputed response. Use a box or ball feasible set Y_t=[c(h_t)-r,c(h_t)+r], with c(h_t) produced by a bounded MLP and r positive and fixed initially. Define a convex inner objective L_inner(y;h_t,x_t)=1/2 y^T A(h_t,x_t)y-b(h_t,x_t)^T y, with A=LL^T+lambda I to guarantee positive definiteness. Solve it with projected gradient: initialize y from the nearest cached item or zero, then repeat y <- Pi_Y(y-alpha(Ay-b)) for K iterations. For each new history, compute a cheap proxy distance d_i between z_t and cached z_i. A stronger version uses Sinkhorn W2 on two short windows of projected latent features; the MVP uses ||z_t-z_i||_2. Reuse y_i if d_i <= tau and ||x_t-x_i||_2 <= tau_x; otherwise solve the inner problem and insert the result into the cache. In the outer model, update x with x <- x + dt*(Pi_X(x-Phi(t,xi_t,x,y))-x), or feed y directly into the predictor and train end-to-end. Calibrate C_W and tau on held-out data by fitting the 95th percentile of response error versus d_i^{1/2}; the Holder exponent comes from the paper, while constants are estimated empirically. The first experiment should use a GRU or 2-layer Transformer on a synthetic nonstationary time-series task with drifting noise distributions. Compare full K-step recomputation against nearest-cache reuse and an unconstrained amortized MLP at equal prediction error. Measure cache hit rate, wall-clock latency, response constraint violations, and forecast loss. Success means at least 5x lower inner-layer compute with less than 1% validation-loss degradation and no sharp error spikes under gradual environment drift.