Uniformly bounded Jacobi spectral features

1. Integration point: implement a bounded Jacobi feature encoder before a transformer, MLP, or graph neural network. Normalize every scalar coordinate u to x=2(u-u_min)/(u_max-u_min)-1, or for graph spectral inputs use x=2lambda/lambda_max-1, where lambda is a Laplacian eigenvalue. Evaluate degrees n=0,...,N and concatenate the features. 2. Pseudocode: choose fixed alpha,beta>=0 and maximum degree N; for each x, compute P_0^(alpha,beta)(x)=1 and evaluate higher Jacobi polynomials with a stable three-term recurrence; multiply each polynomial by the gamma normalization and endpoint factors in the first formula; stack g=[g_0,...,g_N]; form z=Cg; optionally replace C by the l1-normalized C_tilde defined above; pass z to the downstream network. Use Clenshaw evaluation or recurrence in float32 for moderate N, and avoid explicit polynomial expansion. 3. Compute from the paper: the basis normalization and certified bound B_n. Estimate empirically: actual feature maxima, activation variance, gradient norms, and approximation error on held-out inputs. 4. First experiment: train a small CIFAR-10 MLP or a 2D sinusoidal-regression transformer with degree-32 positional features, comparing raw monomials, unnormalized Jacobi polynomials, Fourier features, and normalized Jacobi features. Measure loss at equal FLOPs, maximum activation and gradient, validation error, and NaN frequency while increasing learning rate. Success means lower activation growth and fewer optimization failures at high degree, with equal or better validation accuracy; separately test whether the l1 coefficient constraint permits a larger stable learning rate.