  We briefly review the spin-bit formalism, describing the non-planar dynamics
of the $\mathcal{N}=4,d=4$ Super Yang-Mills SU(N) gauge theory. After
considering its foundations, we apply such a formalism to the $su(2)$ sector of
purely scalar operators. In particular, we report an algorithmic formulation of
a deplanarizing procedure for local operators in the planar gauge theory, used
to obtain planarly-consistent, testable conjectures for the higher-loop $su(2)$
spin-bit Hamiltonians. Finally, we outlook some possible developments and
applications.
