  The calculation of the fundamental limits of nonlinear susceptibilities
posits that when a quantum system has a nonlinear response at the fundamental
limit, only three energy eigenstates contribute to the first and second
hyperpolarizability. This is called the three-level ansatz and is the only
unproven assumption in the theory of fundamental limits. All calculations that
are based on direct solution of the Schrodinger equation yield intrinsic
hyperpolarizabilities less than 0.709 and intrinsic second
hyperpolarizabilities less than 0.6. In this work, we show that relaxing the
three-level ansatz and allowing an arbitrary number of states to contribute
leads to divergence of the optimized intrinsic hyperpolarizability in the limit
of an infinite number of states - what we call the many-state catastrophe. This
is not surprising given that the divergent systems are most likely not
derivable from the Schrodinger equation, yet obey the sum rules. The sums rules
are the second ingredient in limit theory, and apply also to systems with more
general Hamiltonians. These exotic Hamiltonians may not model any real systems
found in nature. Indeed, a class of transition moments and energies that come
form the sum rules do not have a corresponding Hamiltonian that is expressible
in differential form. In this work, we show that the three-level ansatz acts as
a constraint that excludes many of the nonphysical Hamiltonians and prevents
the intrinsic hyperpolarizability from diverging. We argue that this implies
that the true fundamental limit is smaller than previously calculated. Since
the three-level ansatz does not lead to the largest possible nonlinear
response, contrary to its assertion, we propose the intriguing possibility that
the three-level ansatz is true for any system that obeys the Schrodinger
equation, yet this assertion may be unprovable.
