Fortuitous BMN Operators from Multiple Cores

We develop a systematic construction of fortuitous cohomology classes in the BMN sector of $\mathcal N=4$ super Yang--Mills theory from multiple lower-rank building blocks. For arbitrary partitions of colour space, we construct corrected effective letters that obey the supersymmetry transformations of the corresponding lower-rank theories. These allow us to build gauge-invariant, $Q$-closed polynomial operators. A restriction map recovers the constituent cohomology classes, establishing non-exactness and fortuity without brute-force computations. As an explicit example, we combine two fortuitous $SU(2)$ cores to construct a fortuitous class in $SU(4)$. Our results provide a systematic approach to the structure of fortuitous cohomology at finite $N$.