D'Alembert's functional equation f(x+y) + f(x-y) = 2f(x)f(y) characterizes the cosine function: its continuous solutions on the real line are cos(ax) for various frequencies a. The equation captures what it means for a function to have a specific additive-multiplicative relationship between its values at sums and differences.
The inevitability theorem proves that d'Alembert's equation is not one characterization among many but the only possible characterization of a broad class. Start with a general functional equation where the “combiner” — the function that merges f(x+y) and f(x-y) — is allowed to be any polynomial of degree at most 2. Require only that the equation admit continuous, non-constant solutions and that the combiner be symmetric in f(x+y) and f(x-y).
Under these minimal conditions, the equation must be of d'Alembert type. The symmetry condition and the quadratic degree bound together force a unique functional equation structure. No other symmetric quadratic combiner produces well-behaved functional equations. The form is inevitable — not chosen from alternatives but forced by constraints.
The proof proceeds by showing that the constraints progressively narrow the space of possible combiners. Symmetry eliminates odd terms. The degree bound eliminates higher-order terms. The requirement for non-constant solutions eliminates degenerate cases. What remains is precisely d'Alembert. Each constraint removes possibilities; none adds structure. The canonical form is what survives elimination.
The structural lesson: some mathematical objects are unique not because they have special properties but because all alternatives are impossible. D'Alembert's equation is not elegant because someone chose a beautiful form. It is the only form that survives a natural set of constraints. The beauty is the inevitability — the absence of alternatives, not the presence of design.