  We analyze nested Bethe ansatz (NBA) and the corresponding finite size
corrections. We find an integral equation which describes these corrections in
a closed form. As an application we considered the conjectured
Beisert-Staudacher (BS) equations with the Hernandez-Lopez dressing factor
where the finite size corrections should reproduce generic one (worldsheet)
loop computations around any classical superstring motion in the AdS_5 x S^5
background with exponential precision in the large angular momentum of the
string states. Indeed, we show that our integral equation can be interpreted as
a sum over all physical fluctuations and thus prove the complete 1-loop
consistency of the BS equations. In other words we demonstrate that any local
conserved charge (including the AdS Energy) computed from the BS equations is
indeed given at 1-loop by the sum of charges of fluctuations up to
exponentially suppressed contributions. Contrary to all previous studies of
finite size corrections, which were limited to simple configurations inside
rank one subsectors, our treatment is completely general.
