Theorem. Common-scalar flow is a time-reparameterized gradient flow [ftip-00BN]
✍️source

    Let \(\theta (t)\) satisfy Definition [ftip-00BL], let \(t(\tau )\) be the inverse of the clock in Definition [ftip-00BM], and put \(\widetilde \theta (\tau )=\theta (t(\tau ))\). Then, for almost every \(\tau \),

    \[ \widetilde \theta '(\tau )=-\nabla L(\widetilde \theta (\tau )). \]

    Proof. 

      The chain rule for absolutely continuous changes of variable gives \(dt/d\tau =a(t)\) almost everywhere. Multiplying \(\dot \theta =-\nabla L/a(t)\) by \(dt/d\tau \) yields the displayed gradient flow.

      See Theorem 4.6 and its proof in Appendix B.9 of The loss does not see the basis, but Adam does[singh2026lossbasis]. Gauge invariance of \(a\) is not needed for this single-trajectory clock identity; it makes the clock common across a gauge orbit.