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Within the cuprate constellation, one fixed star has been the superconducting dome in the quantum phase diagram of transition temperature vs. the excess charge on the Cu in the CuO2-planes, p, resulting from O-doping or cation substitution. However, a more extensive search of the literature shows that the loss of the superconductivity in favor of a normal Fermi liquid on the overdoped side should not be assumed. Many experimental results from cuprates prepared by high-pressure oxygenation show Tc converging to a fixed value or continuing to slowly increase past the upper limit of the dome of p = 0.26–0.27, up to the maximum amounts of excess oxygen corresponding to p values of 0.3 to > 0.6. These reports have been met with disinterest or disregard. Our review shows that dome-breaking trends for Tc are, in fact, the result of careful, accurate experimental work on a large number of compounds. This behavior most likely mandates a revision of the theoretical basis for high-temperature superconductivity. That excess O atoms located in specific, metastable sites in the crystal, attainable only with extreme O chemical activity under HPO conditions, cause such a radical extension of the superconductivity points to a much more substantial role for the lattice in terms of internal chemistry and bonding.