  The remarkable universality of the cuprate $T_c$ dome suggests a very
fundamental unifying principle. Moreover, the superconducting gap is known to
persist above $T_c$ in the pseudogap phase of all cuprates. So, contrary to
BCS, the gap cannot be the order parameter of the transition.
  In this work, we show that both the $T_c$-dome and the pseudogap line
$T^*(p)$ arise from a unique and identifiable principle: the interaction of
localized `pairons' on an antiferromagnetic square lattice. The topological
constraints on such preformed pairons give rise to both the $T_c$ dome and the
pairing energy {\it simultaneously}. It also provides a natural explanation for
the critical doping points of the phase diagram.
  The model matches perfectly both the $T^*$ and $T_c$ experimental lines, with
only one adjustable parameter.
