  The doping mechanism and realistic Fermi surface (FS) evolution of
La$_{2-x}$Sr$_x$CuO$_4$ (LSCO) are modelled within an extensive ab-initio
framework including advanced band-unfolding techniques. We show that ordinary
Kohn-Sham DFT+U can reproduce the observed metal-insulator transition, when not
restricted to the paramagnetic solution space. Arcs are self-doped by orbital
charge transfer within the Cu-O planes, while the introduced Sr charge is
strongly localized. Arc protection and the inadequacy of the rigid-band picture
are consequences of a rapid change in orbital symmetry at the Fermi energy: the
material undergoes a dimensional crossover along the Fermi surface, between the
nodal (2D) and antinodal (3D) regions. In LSCO, this crossover accounts for FS
arcs, the antinodal pseudogap, and insulating behavior in $c$-axis
conductivity, all ubiquitous phenomena in high-T$_c$ cuprates. Ligand Coulomb
integrals involving out-of-plane sites are principally responsible for the most
striking effects observed by ARPES in LSCO.
