Polynomial-Mixing Block Training

Integrate this mechanism at the data-loader and optimizer level for an RNN, SSM, or world model trained on long trajectories. Choose a binary event e_t=1{g_theta(s_t)>tau}, where g_theta is a constraint score, saturation score, collision score, reward threshold, or gradient-clipping indicator. Partition every trajectory into blocks of length L_n separated by gaps k_n. Use separated blocks as training units, or weight the loss and gradient by an inverse effective sample size estimated from the blocks. For discontinuous events, construct a finite-resolution surrogate tilde_e_t=sigmoid((g_theta(s_t)-tau)/T_n), and lower T_n as the resolution scale increases. Pseudocode: estimate event autocorrelation; set k_n=ceil(k_0 n^epsilon) and L_n=ceil(L_0 n^a); collect separated blocks; compute S_m=sum_{t in I_m}(e_t-p_hat); estimate block variance; normalize event-loss gradients by 1/sqrt(V_block); update parameters; periodically refit the cumulative residual slope on log-log axes. The paper supplies the blocking construction and exponent formula, while xi and alpha must be estimated empirically from autocorrelation decay and return-time tails. First test a small GRU on a synthetic chaotic-map prediction task or Moving-MNIST, comparing contiguous minibatches, shuffled frames, and separated-block training. Measure cumulative threshold-event error versus trajectory length p. The prediction is a residual slope gamma matching the formula within 20 percent; if the slope exceeds 1/2, increase k_n or soften the event approximation. Also test whether block variance predicts validation-loss variance and long-horizon failure frequency better than iid minibatch variance.