  Upon introducing charge carriers into the copper-oxygen sheets of the
enigmatic lamellar cuprates the ground state evolves from an insulator into a
superconductor, and eventually into a seemingly conventional metal (a Fermi
liquid). Much has remained elusive about the nature of this evolution and about
the peculiar metallic state at intermediate hole-carrier concentrations (p).
The planar resistivity of this unconventional metal exhibits a linear
temperature dependence (\rho $\propto$ T) that is disrupted upon cooling toward
the superconducting state by the opening of a partial gap (the pseudogap) on
the Fermi surface. Here we first demonstrate for the quintessential compound
HgBa$_2$CuO$_{4+\delta}$ a dramatic switch from linear to purely quadratic
(Fermi-liquid-like, \rho $\propto$ T$^2$) resistive behavior in the pseudogap
regime. Despite the considerable variation in crystal structures and disorder
among different compounds, our result together with prior work gives new
insight into the p-T phase diagram and reveals the fundamental resistance per
copper-oxygen sheet in both linear (\rho_S = A_{1S} T) and quadratic (\rho_S =
A_{2S} T$^2$) regimes, with A_{1S} $\propto$ A_{2S} $\propto$ 1/p. Theoretical
models can now be benchmarked against this remarkably simple universal
behavior. Deviations from this underlying behavior can be expected to lead to
new insights into the non-universal features exhibited by certain compounds.
