  The static second hyperpolarizability is derived from the space-fractional
Schr\"{o}dinger equation in the particle-centric view. The Thomas-Reiche-Kuhn
sum rule matrix elements and the three-level ansatz determines the maximum
second hyperpolarizability for a space-fractional quantum system. The total
oscillator strength is shown to decrease as the space-fractional parameter
$\alpha$ decreases, which reduces the optical response of a quantum system in
the presence of an external field. This damped response is caused by the
wavefunction dependent position and momentum commutation relation. Although the
maximum response is damped, we show that the one-dimensional quantum harmonic
oscillator is no longer a linear system for $\alpha \neq 1$, where the second
hyperpolarizability becomes negative before ultimately damping to zero at the
lower fractional limit of $\alpha \rightarrow 1/2$.
