Cuprate superconductors have long been known to exhibit an energy gap that persists high above the superconducting transition temperature ($T_c$). Debate has continued now for decades as to whether it is a precursor superconducting gap or a pseudogap arising from some competing correlation. Failure to resolve this has arguably delayed explaining the origins of superconductivity in these highly complex materials. Here we effectively settle the question by calculating a variety of thermodynamic and spectroscopic properties, exploring the effect of a temperature-dependent pair-breaking term in the self-energy in the presence of pairing interactions that persist well above $T_c$. We start by fitting the detailed temperature-dependence of the electronic specific heat and immediately can explain its hitherto puzzling field dependence. Taking this same combination of pairing temperature and pair-breaking scattering we are then able to simultaneously describe in detail the unusual temperature and field dependence of the superfluid density, tunneling, Raman and optical spectra, which otherwise defy explanation in terms a superconducting gap that closes conventionally at $T_c$. These findings demonstrate that the gap above $T_c$ in the overdoped regime likely originates from incoherent superconducting correlations, and is distinct from the competing-order pseudogap that appears at lower doping.
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