Proper-time gauge for a massive worldline einbein ($ds=ed\lambda$)

For the relativistic particle action $S=(m/2)\int [g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }/e-e]d\lambda$, varying the worldline einbein gives $g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }=-e^2$. On the positive-einbein branch the proper time satisfies $ds=ed\lambda$. A reparametrization with ${\lambda }^{\prime }=s$ sets the transformed einbein to one, making the geodesic equation affinely parametrized. This gauge choice is made after retaining the einbein mass shell constraint; fixing the action first would discard that equation. If the parameter interval is held fixed at both ends, its integrated proper length is a modulus and cannot be arbitrarily changed by endpoint-preserving transformations.