Moonshine Uniqueness via the Trace-Product Identity

Let V be a strongly rational holomorphic vertex operator algebra of central charge c = 24 with V₁ = 0. We prove the universal identity Trₕ䃒 (Lₐ Lb Lc) = 900 (a · b | c) for all a, b, c in the orthogonal complement of the Virasoro element in V₂, which we call the Trace-Product Identity, by applying Zhu's n-point recursion. The coefficient gamma₁ = 900 depends only on the partition function J (tau) and the universal data (c = 24, V₁ = 0, dim V₂ = 196884). The full cubic Casimir on V₂ involves two further universal constants (gamma₂, gamma₃) = (620, 744), where gamma₃ coincides with the constant term of the Klein j-function. The identity has two consequences. First, the spectrum of Lₑ at any Ising vector e in V₂ is forced to be the Baby Monster decomposition 2, 0, 1/2, 1/16 with multiplicities (1, 96256, 4371, 96256). Second, a variational argument on the cubic invariant kappa₃ (h) = (h · h | h), using the spectral rigidity above as input, produces 48 mutually orthogonal Ising vectors in V₂. The Dong-Griess-Hohn classification of framed vertex operator algebras then identifies V with the moonshine module V-natural. This proves the Frenkel-Lepowsky-Meurman uniqueness conjecture in its strongly rational form: every strongly rational holomorphic VOA of c = 24 with V₁ = 0 is isomorphic to the moonshine module V-natural.