Slow is fast: raising barriers to accelerate thermal relaxation

What the math gives to ML

The paper identifies a finite-horizon relaxation mechanism that is invisible to instantaneous conductance and eigenvalue comparisons: switching between noncommuting reversible generators rotates relaxation eigenvectors and reprojects residual amplitude into rapidly decaying modes. The optimal control is bang-bang and can transiently raise barriers, so the best finite-time schedule need not use the fastest instantaneous generator. This transfers naturally to optimization by switching among multiple preconditioners, metrics, or optimizer update operators whose induced error dynamics do not commute. The key falsifiable prediction is that gains disappear when the update operators commute, while terminal-loss improvement grows with a measurable commutator and is often obtained by a small number of hard switches rather than smooth interpolation.