  Fluctuation scaling is observed phenomenon from complex networks through
finance to ecology. It means that the variance and the mean of a specific
quantity are related as $\ev{\sigma^2|n}\propto \ev{n|A}^{2\alpha}$ with
$1/2\geq \alpha \geq 1$ when a parameter $A$ (usually the system size) is
varied. $A$ can be the strength of the node, the capitalization of the firm or
the area of the habitat. On the other hand, quantities often obey gap scaling
meaning that their density function depends on, say, the system size $A$ as
$P(n|A) = n^{-1}F(n/A^{\Phi})$. This note describes that these two notions
cannot coexist except when $\alpha = 1$. In this way one can empirically
exclude the possibility of gap scaling in many complex systems including
population dynamics, stock market fluctuations and Internet router traffic,
where $\alpha < 1$.
