A cautionary tale about substituting constraints into Lagrangians

A recent paper claimed to find dynamical vector modes in a modified gravity framework called New General Relativity that earlier work had missed. Here, Golovnev works through the simplest case — vector perturbations in Type 3 New GR — and shows the new claim arose from incorrectly substituting constraint equations directly into the Lagrangian before varying it, a procedure that can spuriously promote non-dynamical modes into apparent degrees of freedom. The lesson matters broadly: counting gravitational degrees of freedom is surprisingly easy to get wrong, with real consequences for which modified gravity theories are considered viable.