Sobolev-optimal frozen analytic features

Integrate this as a frozen first hidden layer for a regression MLP or as a standalone fixed-feature model. Normalize every input coordinate to [-1,1]. Choose an analytic activation with nonvanishing Taylor coefficients, for example sigma(z)=exp(alpha z) with alpha selected so that alpha times the largest projected input remains numerically safe; tanh is also analytic but should be tested for coefficient cancellation and saturation. Generate m scale values with t_k=cos((2k-1)pi/(2m)) and map them to [u_min,u_max]. Generate L quasi-uniform directions v_l on the unit sphere using a spherical Fibonacci construction in low dimensions or normalized low-discrepancy vectors in higher dimensions. Use zero biases initially, then optionally add a second Chebyshev grid of biases. Build Phi[i,(k,l)]=sigma(u_k times dot(v_l,X_i)+b_k), with no gradient through the feature parameters. Solve the output head by A=(Phi^T Phi+lambda I)^(-1)Phi^T Y, using Cholesky for moderate width or conjugate gradients for large width. For classification, replace ridge regression with a softmax linear head. Estimate the theorem's rates empirically by fitting the slope of log validation error against log feature count. First compare against a trainable one-hidden-layer MLP and Gaussian random features on synthetic Sobolev functions formed from Fourier coefficients decaying as |q|^{-(r+d/2)}. Then test MNIST or CIFAR-10. Record wall-clock training time, peak memory, accuracy or MSE, condition number of Phi, and error slopes. Success is lower training cost and a steeper or equal error-width curve than random features at equal feature count, without numerical instability.