  The cuprate superconductors display several characteristic temperatures which
decrease as the material composition is doped, tracing lines across the
temperature-doping phase diagram. Foremost among these is the pseudogap
transition. At a higher temperature a peak is seen in the magnetic
susceptibility, and changes in symmetry and in transport are seen at other
characteristic temperatures. We report a meta-analysis of all measurements of
characteristic temperatures well above $T_c$ in strontium doped lanthanum
cuprate (LSCO) and oxygen doped YBCO. The experimental corpus shows that the
pseudogap line is one of a family of four straight lines which stretches across
the phase diagram from low to high doping, and from $T_c$ up to $700$ K. These
lines all originate from a single point near the overdoped limit of the
superconducting phase and increase as doping is reduced. The slope of the
pseudogap lines is quantized, with the second, third, and fourth lines having
slopes that are respectively $1/2,\;1/3,$ and $1/4$ of the slope of the highest
line. This pattern suggests that the cuprates host a single mother phase
controlled by a 2-D sheet density which is largest at zero doping and which
decreases linearly with hole density, and that the pseudogap lines, charge
density wave order, and superconductivity are all subsidiary effects supported
by the mother phase.
