function describes deviations from the PFA. The width of the roughness spectrum $\sigma[{\bf k}]$ is indeed fixed by the inverse of the correlation length $\ell_{\rm C}$. When this spectrum is contained in the region where $G_{\rm rough}[k]$ remains close to its secular limit $G_{\rm rough}[0]$, we can approximate Eq.(\[beyondPFArug\]) as proportional to the rms roughness $$\begin{aligned}
\delta E_{\rm PP}\simeq G_{\rm rough}[0]\langle h_{1}^{2}+h_{2}^{2}\rangle.  \label{PFArug}\end{aligned}$$ This corresponds effectively to the PFA expression, as the consequence of a theorem which was proved in [@MaiaNetoPRA05] $$\begin{aligned}
G_{\rm rough}[k\rightarrow 0]= \frac{1}{2}\frac{\partial^{2}E_{\rm PP}}{\partial L^{2}}.
\label{PFtheorem}\end{aligned}$$ Equation (\[PFtheorem\]) is nothing but a properly stated “Proximity Force Theorem”. It can however not be confused with the “Proximity Force Approximation” (\[PFArug\]) which is a good approximation only for smooth enough mirrors, that is also for large enough roughness correlation lengths $\ell_{\rm C}$.

In the general case, the PFA result (\[PFArug\]) underestimates the effect of roughness. When performing the theory-comparison, one has therefore to carefully assess the roughness correction by measuring the roughness spectra *in situ* and using the roughness sensitivity function as given in [@MaiaNetoPRA05; @MaiaNetoEPL05]. The PFA can only be used if $\ell_{\rm C}$ has been proven to be large enough or, in a looser way, when the 