the first term gives rise to an effective contact term which is attractive. Thus one cannot factor out a universal distance dependence. In fact eq.(\[ours\]) seems to encapsulate mixed UV and IR effects. This is perhaps not surprising considering the results in section (\[clave\]). The presence of the quartic terms is crucial for the presence of winding states and duality in the emergent dimension.

Before proceeding, some comments about simple potentials are in order:

-   A linear potential $V=a\phi+c$ always verify our SSWGC. This means that the value of $|\phi|$ is unconstrained and may be trans-Planckian with no inconsistency.

-   A pure quadratic potential $V=m^2\phi^2$ is special. In this case the condition is violated for any value of $\phi$ with $m^2>0$. This may be interpreted as a condition that [*forbids the existence of massive scalars with no interaction other than gravity*]{}. It reminds the $U(1)$ WGC which also states that gauge bosons must have at least one charged particle to interact with.

-   For a purely cubic(quartic) potential $V=\mu\phi^3(V=\lambda\phi^4)$ the conditions are fulfilled only if $|\phi|\leq \sqrt{2}M_{\text{p}}(|\phi|\leq \sqrt{6}M_{\text{p}})$.

An exponential potential of the simple runaway form $V=\exp(\pm \alpha\phi)$ passes the test as long as $|\alpha|\geq 1/M_{\text{p}}^2$. Note also that the 