Fractal Uncertainty and Quantitative Uniqueness for the Fourier Bessel Transform

What the math gives to ML

The paper proves a quantitative uncertainty principle for the Fourier–Bessel transform: a function cannot have most of its energy concentrated on a spatial set and simultaneously have its transform concentrated on a sufficiently separated regular frequency set. The transferable asset is the explicit scale-dependent suppression factor h^beta, not the specific Bessel transform. A practical neural-network adaptation is a multiscale spectral regularizer that penalizes hidden states or attention outputs when they are simultaneously localized on irregular token subsets and narrow frequency bands. This is an empirical discrete analogue of the theorem and should be tested primarily for robustness and prevention of representation collapse.