Definition. Common-scalar preconditioned flow [ftip-00BL]
✍️source

    Let \(\theta =(U,V)\) and let \(a(t)>0\) be measurable with \(1/a\) locally integrable. A common-scalar preconditioned flow is a locally absolutely continuous trajectory satisfying

    \[ \dot \theta (t)=-\frac {\nabla L(\theta (t))}{a(t)} \]

    for almost every \(t\). The same scalar multiplies every coordinate of both factor gradients. For gauge-uniform conclusions, the source additionally requires \(a(t)\) to be determined by gauge-invariant statistics of the trajectory up to time \(t\).