  Developing a theory of high-temperature superconductivity in copper oxides is
one of the outstanding problems in physics. Twenty-five years after its
discovery, no consensus on the microscopic theory has been reached despite
tremendous theoretical and experimental efforts. Attempts to understand this
problem are hindered by the subtle interplay among a few mechanisms and the
presence of several nearly degenerate and competing phases in these systems.
Here we provide unified parameter-free explanation of the observed
oxygen-isotope effects on the critical temperature, the magnetic-field
penetration depth, and on the normal-state pseudogap for underdoped cuprate
superconductors within the framework of the bipolaron theory compatible with
the strong Coulomb and Froehlich interactions, and with many other independent
observations in these highly polarizable doped insulators. Remarkably, we also
quantitatively explain measured critical temperatures and magnitudes of the
magnetic-field penetration depth. The present work thus represents an ultimate
proof of the bipolaron theory of high-temperature superconductivity, which
takes into account essential Coulomb and electron-phonon interactions.
