Nested Bandwidth Feature Curriculum

1. Integration point: modify the Fourier-feature embedding before a coordinate MLP or neural operator. Partition frequencies into nested blocks W_1 subset W_2 subset ... , where block k contains frequencies sampled uniformly from Q_(S_k) = [-S_k,S_k]^d. Keep all blocks in memory but gate later blocks off during early training. 2. Pseudocode: estimate the domain radius R, choose delta = 0.05, and select orders J_k = 2^k J_0. Set S_k = (J_k log J_k)^(1/s) / (4 R_*) for ultra-analytic data, or use a known terminal bandwidth S for bandlimited data. Allocate N_k features so cumulative count satisfies N_k >= 6 J_k^d log(28 J_k^d / delta). At epoch e, activate the smallest k allowed by the curriculum schedule, compute Z_k = sqrt(2) cos(X W_k + b_k), concatenate active blocks, and rescale each block by its running RMS. 3. Compute S_k, J_k, and the sample-size targets from the displayed formulas. Estimate unknown R_*, s, and terminal bandwidth from a validation Fourier spectrum. Use lambda_(J,S) as a conditioning diagnostic; when it becomes very small, apply ridge stabilization to the active feature head using G + rho I, with rho proportional to lambda_(J,S) after clipping to a numerically safe interval. 4. First cheap experiment: regress 2D synthetic functions with known Fourier bandwidth on a small MLP using 64 to 1024 total features. Compare the nested curriculum against all features active from step zero and fixed-window RFF at equal total FLOPs. Track loss during the first 20 percent of updates, final test error, gradient-norm spikes, and activation-block utilization. Success means faster early loss descent, fewer optimizer instabilities, and lower high-frequency test error at the same final feature count.