Support Is Free, Bandwidth Is Not: Basis-Parameterised Convolutions for Generative EEG.

Functional convolutions write a kernel as $q$ coefficients over a fixed basis sampled on an $R$-point grid, so their parameter count does not depend on the support $R$. We show that the ratio $R/q$ buying that free support is also a band-limit: a Fourier-basis kernel cannot represent frequencies above $f_s/(2R/q)$, and Legendre and B-spline kernels follow the same $R/q$ scaling (slope 1.00–1.04, $R^2\ge 0.997$) with basis-specific constants. We test what this costs on raw four-channel consumer EEG, with a flow-matching model whose velocity predictor is built from these kernels and held at a 64–128 Hz cutoff against a 40 Hz signal band. Across six checkpoints per model its samples carry 1.21±0.60 times real beta power (13–30 Hz) and 1.63±0.95 times real gamma (30–40 Hz), against 0.052 and 0.020 for a functional VAE on the same substrate, with no overlap in either band ($p=0.00108$). A discrete-kernel twin of the velocity predictor is not distinguishable from it in either band over three paired seeds, so this fidelity is not what basis parameterisation buys. What it buys is an analysable band-limit, and spending that headroom on longer support raises held-out loss monotonically at every basis budget while parameter count explains none of it ($R^2=0.004$). A linear aggregate 0–40 Hz spectral error ranks the VAE better on the same twelve checkpoints and does not separate the two models ($p=0.91$), because two bins below 0.5 Hz account for at least 94% of it. Over 30 seeds, a gaze regressor trained only on the flow model's samples at 5× real volume shows no confirmed degradation against one trained on real data, while the VAE's samples drive it to zero correlation. A 1 Hz high-pass shows that this task reads sub-1 Hz ocular signal. Support is free in parameters; it is paid for in bandwidth.