Endogenous Time in Variational Mechanics: A Scalar Clock with Covariant Dynamics

We propose a covariant upgrade of \emph{external-time weighted} variational principles by introducing an \emph{endogenous scalar clock} \(U\) and a scalar coupling \(W(U)\). The new action\[S[q,U]=\int \!\Big(W(U)\,L(q,\dot q,t)+\frac{\kappa_U}{2}\dot U^2 - V(U)\Big)\dd t\]co-determines ``when things happen'' and ``how they evolve'' at the same variational footing. In a 4D covariant field formulation,\[S[g,U,\psi]=\int \sqrt{-g}\,\Big(\frac{R}{16\pi G}+W(U)\,\cL_m(\psi,\nabla\psi,g)+\frac{\kappa_U}{2}\nabla_\mu U\nabla^\mu U - V(U)\Big)\dd^4x,\]diffeomorphism invariance implies total covariant conservation \(\nabla_\mu(T_m^{\mu\nu}W+T_U^{\mu\nu})=0\). The model strictly contains the classic weighted action with external \(w(t)\) as the degenerate limit \(U\to t\). We develop second- and fourth-order discrete variational integrators (midpoint and GL2), prove corner conditions in the presence of temporal interfaces, and outline observable diagnostics including unit-time windows and bandwise group-delay \(\kappa\). A manufactured-solution test demonstrates near fourth-order convergence for the GL2 scheme.