Moonshine Uniqueness via the Trace-Product Identity

Let V be a strongly rational holomorphic vertex operator algebra of central charge c = 24 with V_1 = 0. We prove the universal identity Tr_{V_2}(L_a L_b L_c) = 900 (a · b | c) for all a, b, c in the orthogonal complement of the Virasoro element in V_2, which we call the Trace-Product Identity, by applying Zhu's n-point recursion. The coefficient gamma_1 = 900 depends only on the partition function J(tau) and the universal data (c = 24, V_1 = 0, dim V_2 = 196884). The full cubic Casimir on V_2 involves two further universal constants (gamma_2, gamma_3) = (620, 744), where gamma_3 coincides with the constant term of the Klein j-function. The identity has two consequences. First, the spectrum of L_e at any Ising vector e in V_2 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 kappa_3(h) = (h · h | h), using the spectral rigidity above as input, produces 48 mutually orthogonal Ising vectors in V_2. 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 V_1 = 0 is isomorphic to the moonshine module V-natural.