In a successful recent experiment on IN5 (ILL), we have been able to show that in contrast to commonly accepted wisdom, the excitation gap in nearly optimally doped La(1.855)Sr(0.145)CuO(4) does not open at the superconducting transition temperature, Tc, but at a much lower temperature T0~Tc/2. Further measurements, as well as published data, suggest a highly intriguing trend - consistent with quantum criticality near optimal doping in a simple cuprate superconductor - of an energy scale T0 approaching zero at the critical doping level xc=0.13 at which spin-density wave ordering sets in. In this proposal, we aim to pursue this novel discovery by studying in detail the temperature dependence of the low-energy excitation spectrum of LSCO x=0.16.