proposition. Gauge equivariance gives parameterization stability [ftip-00BV]
✍️source

    Every gauge-equivariant optimizer is parameterization-stable on each orthogonal orbit, provided the remaining protocol coordinates are coupled as in Definition [ftip-00BU].

    Proof. 

      Definition [ftip-00B3] gives \((\widetilde U_t,\widetilde V_t)=(U_tQ,V_tQ)\). Hence

      \[ \widetilde U_t\widetilde V_t^{\mathsf T} =U_tQQ^{\mathsf T}V_t^{\mathsf T}=U_tV_t^{\mathsf T}. \]

      This is a sufficient condition on a declared orbit, not a claim that equivariance is necessary for every possible equality of represented trajectories.