  Understanding the mechanism by which d-wave superconductivity in the cuprates
emerges and is optimized by doping the Mott insulator is one of the major
outstanding problems in condensed matter physics. Our high-resolution scanning
tunneling microscopy measurements of the high transition temperature (Tc)
superconductor Bi2Sr2CaCu2O8+d show that samples with different Tcs in the low
doping regime follow a remarkably universal d-wave low energy excitation
spectrum, indicating a doping independent nodal gap. We demonstrate that Tc
instead correlates with the fraction of the Fermi surface over which the
samples exhibit the universal spectrum. Optimal Tc is achieved when all parts
of the Fermi surface follow this universal behavior. Increasing temperature
above Tc turns the universal spectrum into an arc of gapless excitations, while
overdoping breaks down the universal nodal behavior.
