d'Alembert's Functional Equation on Arbitrary Groups: An Elementary Trace-Algebra Classification, Machine-Checked

We classify the complex solutions of d'Alembert's functional equation f(xy) + f(xy⁻¹) = 2f(x)f(y) on an arbitrary group G, assuming nothing beyond the equation: no commutativity of G, no Kannappan condition, no regularity. Every nonzero solution is the normalized trace f = ½ tr ρ of a semisimple representation ρ : G → SL(2, ℂ), and exactly one of two cases holds: f is a symmetrized multiplicative character (χ + χ⁻¹)/2, which happens precisely when Kannappan's condition f(xyz) = f(xzy) holds, or ρ is irreducible, which happens precisely when that condition fails. The proof is elementary: it associates to f a canonical quotient B_f of the group algebra ℂ[G] carrying a nondegenerate trace pairing and a quadratic Cayley-Hamilton identity, and proves that B_f is ℂ, ℂ × ℂ, or M₂(ℂ). Kannappan's condition is shown to be exactly commutativity of B_f. The classification statement is Davison's theorem; the contribution here is the self-contained trace-algebra route and its formalization. The main exclusive classification, the trace-algebra construction, the converse, the construction part of the three-algebra lemma, and the stated results for Q₈ and Q_{4n} are machine-checked in Lean 4. The introduction records the exact boundary of that claim.