  The basal area (Cu-Cu grid) of the cuprate superconductors not only tends to
shrink on hole doping, as expected from single electron quantum chemistry, but
exhibits also an electronically incompressible "hump'' around optimum doping
n_opt = 0.16. The hump collapses near critical doping n_crit = 0.19. We analyze
the origin of the hump in terms of a classical liquid of interacting
incompressible particles in a container with antiferromagnetic walls. Oxygen
holes interacting with the wall form singlets, protect themselves against other
holes by an incompressible "spin fence'', and thus interact also with the
lattice. Occupation of the CuO_2 lattice with holes must therefore follow a
non-double-occupant constraint also for the oxygen cage enclosing the copper
hole. Closest packing of self-protecting singlets is found to occur around
critical doping; closest packing of paired self-protecting singlets around
optimum doping. These singlet-states are bosonic, but are not magnetic
polarons.
