  The pairing gap plays a fundamental role in the nuclear many-body problem and
many large scale and accurate mass formula fits suggest the smooth nuclear mass
dependence $\Delta \sim 6(1)~ A^{-1/3}~{\rm MeV}$ in the liquid drop model
which lacks a theoretical motivation. Within the BCS theory we analyze the
impact of phase equivalent interactions on the pairing gap for a translational
invariant many-fermion system such as nuclear and neutron matter. To that end
we use explicitly the Similarity Renormalization Group (SRG) transformations.
We show that in the on-shell and continuum limits the pairing gap vanishes. For
finite size systems the pairing gap can be computed directly from the
scattering phase-shifts by the formula $$ \Delta_{nn} (p_F) = \Delta \epsilon_F
~ \delta^{^1S_0}_{nn}(p_F) /\pi ~ , $$ where $p_F$ is the Fermi momentum and
$\Delta \epsilon_F$ the level spacing at the Fermi energy which for the
harmonic oscillator shell model becomes $\Delta \epsilon_F= \hbar \omega \sim
41 ~ A^{-1/3}~{\rm MeV}$, so that $$ \Delta_{nn} (p_F) \sim 4 ~ A^{-1/3}~{\rm
MeV} ~ . $$ The comparison with double differences from binding energies of
stable nuclei is satisfactory and the discrepancy with the large scale analysis
may be attributed to the lack of three-body forces. Nevertheless, the on-shell
two-body interaction provides a basis for the $c~A^{-1/3}$ dependency and
accounts for 75\% of the coefficient $c$.
