The Calibrated d’Alembert Cost on Positive Paths: Convexity Under Step Evaluation

Source abstract

A calibrated solution of d’Alembert’s equation gives the cost K(v)=coshv−1. Under the explicit postulate that this cost is evaluated at log-velocity, we study the resulting free action on positive paths. Geometric interpolation makes the action strongly convex, and Jensen’s inequality gives its unique fixed-endpoint minimizer. An exact Bregman identity converts excess action into a trajectory-error bound. The purpose is quantitative rather than formal: because the minimum action is available in closed form, the excess action of any computed path is itself computable and returns a rigorous bound on that path’s error. We determine the optimal global velocity-gap constant at fixed endpoints and its sharper small-perturbation limit. Sine and tent competitors separate velocity-gap sharpness from equality in the trajectory inequality. Reproducible numerical examples illustrate these estimates and their use as error certificates. A separate scalar-mechanics interpretation yields the Newtonian limit and a cosh-dual Hamiltonian distinct from the special-relativistic Hamiltonian. For the native oscillator, an explicit mixed-harmonic perturbation resolves the degenerate conjugate-time case.