The trace-free Beurling--Ahlfors transform and the Bourgain--Brezis problem for Hodge systems

What the math gives to ML

The paper constructs a canonical trace-free Beurling--Ahlfors transform on differential forms by combining complementary Hodge projections with coefficients chosen to cancel the channel trace. Its useful transferable asset is not the endpoint Bourgain--Brezis theorem itself, which is difficult to encode in a network, but the explicit frequency-wise orthogonal decomposition and reflection-induced cancellation. A neural layer can treat groups of feature channels as discrete differential-form components and apply this zero-parameter Fourier multiplier as a stable structured mixer. The main falsifiable hypothesis is that this constrained mixing improves optimization stability or sample efficiency relative to an equally expensive unconstrained channel mixer.