A Multiplicative Horn Obstruction to the Unitary Categorification of a Rank-Twelve Modular Datum

Abstract

We study the rank-twelve potential modular datum denoted $12^{39,406}_{4,1276.}$ in the low-rank classification tables. Its global dimension is $(1287+351\sqrt{13})/2$, and its fusion ring agrees with that of the known rank-twelve Haagerup modular category. The datum satisfies the modular relations, has nonnegative integral Verlinde coefficients, and passes several familiar genus-one necessary tests. We prove, nevertheless, that it cannot be realized by a unitary modular tensor category. The obstruction already occurs in the representation of the three-strand braid group on $\operatorname{Hom}(C_{[10]},B^{\otimes3})$. Generalized Frobenius--Schur indicators determine the eigenvalues of the two-strand generator on this twelve-dimensional space. The central relation in $B_3$ then reduces the problem to a product of three prescribed unitary conjugacy classes. A sixfold symmetry reduces the $1302$ determinant-compatible cases to $217$ representatives. Rank-one inequalities leave only $17$ of these, and ten explicitly displayed degree-zero rank-two Schubert inequalities exclude the remaining cases. A violated inequality with gap $2/117$ is worked out explicitly. All calculations used in the obstruction are exact. The argument uses unitarity essentially and leaves non-unitary categorification open.