The Berry--Esseen Estimate in the Free Central Limit Theorem

What the math gives to ML

The paper gives a quantitative finite-sample bound showing that a normalized free sum of centered self-adjoint variables is close to the standard semicircle law, with error controlled by the aggregate normalized (2+δ)-moment. The transferable asset is an explicit tail-sensitive certificate: additive operators with bounded higher moments should have predictable aggregate spectra and fewer extreme spectral outliers. This suggests a spectral initialization and regularization rule for architectures built from sums of residual, routing, or attention operators. Neural-network matrices are not exactly freely independent, so the proposed use is an empirical approximate diagnostic rather than a universal theorem.