  In the context of quantum gravitational systems, we place bounds on regions
in field space with slowly varying positive potentials. Using the fact that
$V<\Lambda_s^2$, where $\Lambda_s(\phi)$ is the species scale, and the emergent
string conjecture, we show this places a bound on the maximum diameter of such
regions in field space: $\Delta \phi \leq a \log(1/V) +b$ in Planck units,
where $a\leq \sqrt{(d-1)(d-2)}$, and $b$ is an $\mathcal{O}(1)$ number and
expected to be negative. The coefficient of the logarithmic term has previously
been derived using TCC, providing further confirmation. For type II string flux
compactifications on Calabi--Yau threefolds, using the recent results on the
moduli dependence of the species scale, we can check the above relation and
determine the constant $b$, which we verify is $\mathcal{O}(1)$ and negative in
all the examples we studied.
