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

We study the kinetic action that d’Alembert’s functional equation induces on positive paths in $\text{Rplus}$, and prove it strongly convex. Calibrated d’Alembert forces the cosh cost $\text{Jcost}(x)=\frac{1}{2}(x+x^{-1})-1$, i.e.\ $\text{Jlog}(\xi )=\cosh \xi -1$ 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 $\text{actionA}[\gamma ]={\int }_a^b(\cosh \dot{\xi }-1)dt$, 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 $\text{actionA}[\gamma ]-\text{actionA}[{\gamma }_{\ast }]=\int D_{\text{Kkin}}(\dot{\xi }\mathrm{\parallel }{\dot{\xi }}_{\ast })dt$, sharpened by a quantitative Friedrichs–Poincar'e bound on $\log (\gamma \mathrm{/}{\gamma }_{\ast })$. 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 ${\text{Kkin}}_m(\varphi )=m({\gamma }_L-1)$; 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.