The D’Alembert Inevitability Theorem

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 degP(u,v)≥3 do not admit nonconstant continuous solutions, provided the leading term does not cancel (Theorem 1). For continuous nonconstant functions F:R>0→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+cuv, c∈R (Theorem 3). The equation reduces in logarithmic coordinates to the classical d’Alembert functional equation. For c≠0, one obtains hyperbolic or trigonometric branches, while c=0 yields the squared-logarithm family. Under the cost-function assumptions F≥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)=12(x+x−1)−1. For c≠0, the result extends to R>0n: every solution depends only on a single linear combination of coordinate logarithms; for c=0, the solution is a general quadratic form ∑i,jaijlnxilnxj. In either case, nontrivial coordinate-wise separable costs are excluded.