constraint is insensitive to $V'$ and $V$ themselves so insensitive to whether the theory is in dS or AdS and the conditions for minima. So the constraints here discussed seem unrelated to the dS conditions of ref.[@dS1; @dS2; @krishnan; @dS3; @masdS]. In particular our condition is compatible but independent from the dS conjecture.

The above constraint may be easily generalized to the case of several scalars fields.

A first test: the axion potential
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If the SSWGC applies to any scalars, it should apply also to axions and their periodic potentials which we know appear in string theory whenever an axion-like scalar couples to a non-Abelian gauge group. So one may consider the axion example as a test for the conjecture. The leading instanton contribution to the axion potential has the form V =  - (/f))  . In this case the SSWGC gives (2 \^2(/M\_) +\^2(/M\_))    . Here we have constrained ourselves to the region in which $V''\geq 0$ in which the leading cosine instanton term is expected to dominate. The above expression yields f\^2   M\_\^2(1+2 \^2(/M\_)) and, since the bound must be true for all $\phi$, one obtains $f\leq M_{\text{p}}$. So we see we can derive from the SSWGC 