Therefore, we only need one and one-half loops to find the two-loop dilatation operator. More precisely, using a subscript $n$ for the $\mathcal{O}(g^n)$ corrections, $${{{\textstyle\frac{1}{2}}}}{\delta{\mathfrak{D}}}_4= {\left\{\overrightarrow{{{\mathfrak{T}}}_3},\overleftarrow{{{\mathfrak{T}}}_1}\right\}}+{\left\{\overrightarrow{{{\mathfrak{T}}}_1},\overleftarrow{{{\mathfrak{T}}}_3}\right\}}.$$

The $\mathcal{O}(g^3)$ solution is built only from the leading (non-vanishing) order representations of the generators and the auxiliary generator, ${{\mathfrak{h}}}$, that acts using the harmonic numbers. Schematically, we have[^3] $$\begin{gathered}
 {{\mathfrak{T}}}_3 = \pm {\left[{{\mathfrak{T}}}_1,{{\mathfrak{x}}}\right]}, \notag \\ {{\mathfrak{x}}}\sim {\left\{\overrightarrow{{{\mathfrak{T}}}_1},{\left[\overleftarrow{{{\mathfrak{T}}}_1},{{\mathfrak{h}}}\right]}\right\}}. \end{gathered}$$ It follows that ${\delta{\mathfrak{D}}}_4$ is built only from the ${{\mathfrak{T}}}_1$’s and ${{\mathfrak{h}}}$.

Our solution lifts consistently and naturally to non-planar $\mathcal{N}=4$ gauge theory as well[^4], that is for any choice of the gauge group. In particular, it includes wrapping interactions. These are non-planar contributions to the dilatation generator that survive the planar limit by wrapping around short operators [@Beisert:2004hm]. Because supersymmetry relates shorter states to longer states, wrapping interactions do not contribute until four loops [@Beisert:2003ys; @Beisert:2005fw]. However, generalizing this solution for wrapping interactions to higher loops would provide a missing piece for the comparisons between gauge and string theory. The proposed Bethe ansätze are oblivious to corrections from wrapping interactions[^5].

We also find compelling evidence that integrability persists at two loops in non-compact sectors. The two-loop dilatation generator generates the same bosonic ${{\mathfrak{sl}}}(2)$ subsector 