  This letter calculates the exact fundamental limits of the first ({\beta})
and second ({\gamma}) hyperpolarizabilities using Monte-Carlo sampling of
energy spectra and transition moments constrained by the diagonal
Thomas-Reiche-Kuhn (TRK) sum rules and filtered by the off-diagonal TRK sum
rules. The upper bounds of {\beta} and {\gamma} are determined from these
quantities by applying error-refined extrapolation to perfect compliance to the
sum rules. The method yields the largest diagonal component of the
hyperpolarizabilities for an arbitrary number of interacting electrons in any
number of dimensions without the necessity for exotic interactions or
unconventional spectra. The results show that the fundamental limits agree with
those achieved by potential optimization and direct solution of mechanical
Hamiltonians, about 30% (40%) below original estimates of the limits for
{\beta} ({\gamma}), and show that the special cases that lead to divergent
nonlinearities in the many-state catastrophe are not physically realizable.
Thus, this letter shows for the first time that the fundamental limits are
rigorously exact and are achievable without exotic material designs.
