AI 中文总结
研究达朗贝尔函数方程在正路径上诱导的动力学作用，通过在对数速度处评估对数代价得到强凸作用，其有三个结果，该定理纯数学，与牛顿和快度力学联系有条件，添加凸势后经典图景回归。
AI 中文摘要
我们研究了达朗贝尔函数方程在\(\mathbb{R}_{+}\)中正路径上诱导的动力学作用，并证明其为强凸的。校准后的达朗贝尔力导致cosh代价\(\mathcal{J}_{cost}(x)=\frac{1}{2}(x + x^{-1}) - 1\)，即在对数坐标\(\xi = \log x\)中为\(\mathcal{J}_{\log}(\xi)=\cosh\xi - 1\)。在对数速度\(\dot{\xi}\)而非对数位置处评估此对数代价（一个单一假设（假设\(\ref{post:step}\)）），得到\(\mathcal{A}[\gamma]=\int_{a}^{b}(\cosh\dot{\xi}-1)dt\)，在几何（对数空间）插值下是强凸的。这种凸性有三个结果，无需欧拉 - 拉格朗日方程、弗雷歇导数或二阶变分。一是单边弦条件刻画全局极小性；二是唯一的固定端点极小值是均匀对数速度路径；三是作用间隙服从精确的布雷格曼/毕达哥拉斯恒等式\(\mathcal{A}[\gamma] - \mathcal{A}[\gamma_{*}]=\int D_{\mathcal{K}_{kin}}(\dot{\xi}\|\dot{\xi}_{*})dt\)（通过对\(\log(\gamma / \gamma_{*})\)的定量弗里德里希斯 - 庞加莱界得到加强）。在加法坐标\(\xi\)中有对偶平坦/黑塞流形的解读。该定理是纯数学的，与牛顿力学和快度力学的联系是有条件的，需要假设\(\ref{post:step}\)之外的结构：运动学嵌入、质量耦合、时间校准和哈密顿 - 原初勒让德结构。具备这些条件后，cosh作用恢复牛顿小步极限和快度分布\(\mathcal{K}_{kin,m}(\phi)=m(\gamma_{L}-1)\)；然而cosh对偶哈密顿量不是狭义相对论自由粒子哈密顿量（命题\(\ref{prop:not - SR}\)），一致性在于分布，而非哈密顿量的恒等。全局极小性是自由扇区现象：一旦添加非仿射严格凸势，联合凸性就会丧失，经典的平稳作用图景就会回归。
英文摘要
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.\ $\Jlog(ξ)=\coshξ-1$ in the log coordinate $ξ=\log x$. Evaluating this log-cost at the log-\emph{velocity} $\dotξ$ rather than the log-position -- a single postulate (Postulate~\ref{post:step}) -- yields $\actionA[γ]=\int_a^b(\cosh\dotξ-1)\,dt$, strongly convex under geometric (log-space) interpolation. This convexity has three consequences, none requiring an Euler--Lagrange equation, a Fréchet 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 $\actionA[γ]-\actionA[γ_*]=\int D_\Kkin(\dotξ\,\|\,\dotξ_*)\,dt$, sharpened by a quantitative Friedrichs--Poincaré bound on $\log(γ/γ_*)$. It has a dually-flat / Hessian-manifold reading in the additive coordinate $ξ$. \\ 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 $\Kkin_m(ϕ)=m(γ_L-1)$; 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.
Comments36 pages
