Lag-Spectrum Online Newton

Integrate this into the coefficient-learning part of a causal convolution, FIR filter, SSM input filter, or lightweight RNN whose preactivation is z_t = sum_j theta[j] times U[t+1-j]. Keep the hidden architecture unchanged and compare its coefficient optimizer against AdamW and ordinary diagonal AdaGrad. Inputs are a lag buffer U, target y, envelope values r[j], horizon T, learning rate eta, and epsilon. Initialize theta[j] = 0 and H[j] = 1/(n[j] times r[j]^2 + epsilon), with n[j] = max(T-j+1, 1). At each step t, form z as the sum over active lags, p = sigmoid(z), compute g[j] = (p-y) times U_lag[j], and add h[j] = p times (1-p) times U_lag[j]^2 + 1/(n[j] times r[j]^2 + epsilon) to H[j]. Update theta[j] by the displayed clipped Newton step; inactive lags j greater than t receive no update. If T is unknown, replace n[j] by a running count c[j] of times lag j has been active, and use c[j]+1 in the prior term. The paper supplies the lag complexity and motivates the horizon weighting; the Hessian diagonal and clipping are computable directly, while r[j] can be fixed from an assumed exponential or polynomial envelope or estimated from a pilot run. First test on synthetic binary data generated by exponential and polynomial filters, then on an online event-stream benchmark. Measure cumulative excess log loss, loss versus optimizer steps and FLOPs, coefficient recovery by lag, and stability under long contexts. Success means lower regret at equal compute, especially for old lags, without increasing peak memory.