  We compute the anomalous dimensions of field strength operators Tr F^L in N=4
SYM from an asymptotic nested Bethe ansatz to all-loop order. Starting from the
exact solution of the one-loop problem at arbitrary L, we derive a single
effective integral equation for the thermodynamic limit of these dimensions. We
also include the recently proposed phase factor for the S-matrix of the planar
AdS/CFT system. The terms in the effective equation corresponding to,
respectively, the nesting and the dressing are structurally very similar. This
hints at the physical origin of the dressing phase, which we conjecture to
arise from the hidden presence of infinitely many auxiliary Bethe roots
describing a non-trivial "filled" structure of the theory's BPS vacuum. We
finally show that the mechanism for creating effective nesting/dressing kernels
is quite generic by also deriving the integral equation for the all-loop
dimension of a certain one-loop so(6) singlet state.
