$g$ of the complete theory, in which case as $g^2\rightarrow 0$ a tower of states become massless to preclude the presence of global symmetries in the effective theory.

The argument goes also in the opposite direction. Consider a 4D theory obtained upon compactification of a 5D string theory on a circle. Then the masses of the particles in the KK and winding towers depend on the radion in such a way that their potential verifies the differential equation (\[guay\]). This gives support to the proposed conjecture and the equation.

Given the above discussion, we propose a second conjecture:

Note that the structure is analogous to how the 11-th dimension appears from Type IIA string theory at strong coupling. The analogue of the above extremal states are the towers of $D0$-branes of string theory building up the KK modes of the 11-th dimension. Thus the simplest system one can think of with a single scalar coupled to quantum gravity secretly has several features characteristic of string theory.

Extremal potentials
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Independently of the existence of these towers of states, it is interesting the question of whether one can write down potentials saturating the bound. By integrating $V''$ one can obtain general 