Sharp Complete Modified Log-Sobolev Inequalities on Classical and Quantum Tori

What the math gives to ML

The paper supplies a sharp modified logarithmic Sobolev inequality on the circle and extends it completely to positive matrix-valued functions, with tensorization to classical and quantum tori. The transferable asset is a dimension-independent entropy-to-dissipation guarantee that remains valid when each spatial or token location carries a coupled positive matrix rather than a scalar probability. A practical neural adaptation is to represent attention or routing distributions as positive matrix-valued fields over a periodic index, apply a controllable heat-semigroup smoothing step, and regularize the resulting BKM Fisher information. The predicted benefit is stable local smoothing with a mathematically calibrated time parameter instead of an arbitrary entropy penalty.