  More than three decades after the discovery of superconductivity (SC) in the
cuprates, the nature of the "normal" state and the mechanism of SC remain
mysterious. One popular theoretical approach has been to treat the CuO$_2$
layer as coupled two-leg one-band Hubbard ladders. In the undoped two-leg
ladder spin-singlets occupy ladder rungs, and doped ladders are characterized
by superconducting correlations with quasi-long range order (quasi-LRO).
Pressure-induced SC in Sr$_{14-x}$Ca$_x$Cu$_{24}$O$_{41}$ (SCCO) has long been
explained within one-band ladder theories. The dramatic pressure-driven
crossover from quasi one-to-two dimensional (1D-to-2D) transport and the
simultaneous vanishing of the spin gap due to ladder singlets in the metallic
state preceding SC however lie outside the scope of ladder-based theories.
Recent demonstration of rapid decay of superconducting correlations with
distance within a realistic multiband model of the cuprate ladder gives
additional credence to this viewpoint. Here we show that the assumption of
pressure-driven increase in hole concentration in the ladders cannot explain
the dimensionality crossover. The dimensionality crossover is due to discrete
change in Cu-ion ionicity accompanied by transfer of holes from the Cu to
O-ions, leading to "negative charge-transfer gap". Our theory of valence
transition driven dimensionality crossover in SCCO provides a generic
explanation of the very large increase in charge carrier density at critical
doping concentrations in both hole and electron doped layered cuprates We
propose a falsifiable experimental test of our theory.
