Meromorphic cosets and the classification of three-character CFT

We investigate the admissible vector-valued modular forms having three independent characters and vanishing Wronskian index determine which ones correspond to genuine 2d conformal field theories. This is done by finding bilinear coset-type relations that pair them into meromorphic with central charges 8, 16, 24, 32 40. Such pairings allow us identify some definite CFTs rule out others. As a key result we classify all unitary three-character CFT index, excluding $c=8,16$. The complete list has two infinite affine series $B_{r,1},D_{r,1}$ 45 additional by-product, at higher values of total charge also find constraints on existence or otherwise separately several cases potentially Intermediate Vertex Operator Algebras.