The d'Alembert Inevitability Theorem

Abstract

We study functions satisfying the composition law $F(xy)+F(x/y)=P(F(x),F(y))$ with a symmetric polynomial combiner $P$. We prove that symmetry together with a quadratic degree bound on $P$ forces a composition law of d'Alembert type. We establish a degree mismatch exclusion criterion showing that symmetric polynomial combiners with $\mbox{deg} P(u,v) \ge 3$ do not admit nonconstant continuous solutions, provided the leading term does not cancel (Theorem 3.1.). For continuous nonconstant functions $F:\mathbb{R}_{>0}\to\mathbb{R}$ with $F(1)=0$ satisfying the composition law with a symmetric polynomial $P$ of degree at most two, the combiner is necessarily of the form $P(u,v)=2u+2v+c\,uv$, $c\in\mathbb{R}$ (Theorem 3.3.). The equation reduces in logarithmic coordinates to the classical d'Alembert functional equation. For $c\neq 0$, one obtains hyperbolic or trigonometric branches, while $c=0$ yields the squared-logarithm family. Under the cost-function assumptions $F\ge 0$ and convexity, only the hyperbolic branch with $c>0$ remains. A unit log-curvature calibration selects the canonical value $c=2$, which yields the canonical reciprocal cost $F(x)=\tfrac12(x+x^{-1})-1$. For $c\neq0$, the result extends to $\mathbb{R}_{>0}^n$: every solution depends only on a single linear combination of coordinate logarithms; for $c=0$, the solution is a general quadratic form $\sum_{i,j}a_{ij}\ln x_i\ln x_j$. In either case, nontrivial coordinate-wise separable costs are excluded.