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In the context of quantum gravitational systems, we place bounds on regionsin 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 emergentstring conjecture, we show this places a bound on the maximum diameter of suchregions 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 andexpected to be negative. The coefficient of the logarithmic term has previouslybeen derived using TCC, providing further confirmation. For type II string fluxcompactifications on Calabi--Yau threefolds, using the recent results on themoduli dependence of the species scale, we can check the above relation anddetermine the constant $b$, which we verify is $\mathcal{O}(1)$ and negative inall the examples we studied.

Bounds on Field Range for Slowly Varying Positive Potentials