d'Alembert's Functional Equation and a Globally Convex Free-Action Principle on Positive Paths

Abstract

We study the kinetic action that d'Alembert's functional equation induces on positive paths in $\Rplus$, and prove it strongly convex. Calibrated d'Alembert forces the cosh cost $\Jcost(x)=\tfrac12(x+x^{-1})-1$, i.e.\ $\Jlog(\xi)=\cosh\xi-1$ in the log coordinate $\xi=\log x$. Evaluating this log-cost at the log-\emph{velocity} $\dot\xi$ rather than the log-position -- a single postulate (Postulate~\ref{post:step}) -- yields $\actionA[\gamma]=\int_a^b(\cosh\dot\xi-1)\,dt$, strongly convex under geometric (log-space) interpolation. This convexity has three consequences, none requiring an Euler--Lagrange equation, a Fr\'echet derivative, or a second variation. First, a one-sided chord condition characterizes global minimality. Second, the unique fixed-endpoint minimizer is the uniform-log-velocity path. Third, the action gap obeys an exact Bregman / Pythagorean identity $\actionA[\gamma]-\actionA[\gamma_*]=\int D_\Kkin(\dot\xi\,\|\,\dot\xi_*)\,dt$, sharpened by a quantitative Friedrichs--Poincar\'e bound on $\log(\gamma/\gamma_*)$. It has a dually-flat / Hessian-manifold reading in the additive coordinate $\xi$. \\ This theorem is purely mathematical, and we delimit it. The bridge to Newtonian and rapidity mechanics is \emph{conditional}, requiring structure beyond Postulate~\ref{post:step}: a kinematic embedding, a mass coupling, a time calibration, and a Hamiltonian-primary Legendre structure. Granted these, the cosh action recovers the Newtonian small-step limit and the rapidity profile $\Kkin_m(\phi)=m(\gamma_L-1)$; yet the cosh-dual Hamiltonian is \emph{not} the special-relativistic free-particle Hamiltonian (Proposition~\ref{prop:not-SR}), the agreement being one of profile, not an identity of Hamiltonians. Global minimality is a free-sector phenomenon: once a non-affine strictly convex potential is added, joint convexity is lost and the classical stationary-action picture returns.