proof of this ansatz assuming integrability was found recently [@Beisert:2005tm]. Parallel developments have occurred on the string side. Classical string theory in $AdS_5 \times S^5$ was shown to be integrable [@Mandal:2002fs; @Bena:2003wd], integrability was used to solve the classical spectrum in terms of algebraic curves [@Kazakov:2004qf; @Beisert:2005bm], and Bethe ansätze for quantum strings were proposed [@Arutyunov:2004vx; @Staudacher:2004tk; @Beisert:2005fw].

Despite all this progress and much evidence, there is no rigorous proof of higher-loop integrability for any non-compact sector of the gauge theory. Finding the dilatation generator would be a first step towards a proof. However, even that step seems intractable for non-compact sectors. Completing a direct diagrammatic calculation is realistic only at low loop order, as was done at two loops for the fermionic ${{\mathfrak{sl}}}(2)$ sector in [@Belitsky:2005bu]. A more promising approach to higher loops is Beisert’s method, which takes full advantage of superconformal symmetry. However, extending Beisert’s method of computing the dilatation generator from one loop for the complete theory [@Beisert:2003jj; @Beisert:2004ry] to three loops for the ${{\mathfrak{su}}}(2|3)$ sector [@Beisert:2003ys], depended on compactness. A large computer algebra computation was essential. Because the dilatation generator in non-compact sectors is built from infinitely many interactions, a brute force computation of this kind 