wrapping interactions that only change the anomalous dimensions of two-site states would be inconsistent with the symmetry constraints.

Tests and applications of the solution \[sec:checks\]
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Using the solution for the two-loop dilatation operator, we first provide strong evidence that it is correct via direct diagonalization and comparison to rigorous field theory computations. We then use our solution to present strong evidence in favor of integrability by computing the internal S-matrix in the bosonic ${{\mathfrak{sl}}}(2)$ sector and by comparing anomalous dimension predictions of the Bethe ansatz of [@Staudacher:2004tk] with the results of direct diagonalization.

Two-loop planar anomalous dimensions
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Expanding the expression for ${\delta{\mathfrak{D}}}_4$ in terms of interactions, we find the planar anomalous dimensions by direct diagonalization. We first identify the spin chain states of the subspaces of certain (small) values of classical dimension, R-charge, length, and hypercharge. Then we apply $g^2 {\delta{\mathfrak{D}}}_2 + g^4{\delta{\mathfrak{D}}}_4$ to these subspaces and compute its eigenvalues (the anomalous dimensions) and eigenstates. Again, we have used `Mathematica`. We check states with rigorously known anomalous dimensions. These include twist-two operators [@Kotikov:2003fb], a pair of states of length three and bare dimension six [@Eden:2005bt], two excitation states (BMN operators) [@Beisert:2002tn; @Beisert:2003tq], and length-three states built from one 