  We show that an electronic phase separation (EPS) transition described by the
an appropriate theory yields regions of low free energy forming grains of low
and high charge densities. These local differences in the potential energy are
studied numerically and used here for the first time as the origin of an
attractive interaction that gives rise to local pair formation. The resistivity
transition $T_c$ occurs due to Josephson coupling among these superconducting
regions or grains. Using this approach within the Bogoliubov-deGennes
calculations we derive the Gaussian shape of $T_c$ against the entire doping
level $p$. We find that every pairing gap develops locally at a temperature
$T_p$, following the relation $2\Delta/k_BT_p\approx8.0$ in close agreement
recent measurements.}
