We study the origin and properties of extra or excess central light in the surface brightness profiles of cusp or power-law ellipticals. Dissipational mergers give rise to two-component profiles: an outer profile established by violent relaxation acting on stars present in the progenitors prior to the final merger, and an inner stellar population comprising the extra light, formed in a compact starburst. Combining a large set of hydrodynamical simulations with data that span a broad range of profiles and masses, we show that this picture is borne out -- cusp ellipticals are indeed extra light ellipticals -- and examine how the properties of this component scale with global galaxy properties. We show how to robustly separate the extra light, and demonstrate that observed cusps are reliable tracers of the degree of dissipation in the spheroid-forming merger. We show that the typical degree of dissipation is a strong function of stellar mass, tracing observed disk gas fractions at each mass. We demonstrate a correlation between extra light content and effective radius at fixed mass: systems with more dissipation are more compact. The outer shape of the light profile does not depend on mass, with a mean outer Sersic index ~2.5. We explore how this relates to shapes, kinematics, and stellar population gradients. Simulations with the gas content needed to match observed profiles also reproduce observed age, metallicity, and color gradients, and we show how these can be used as tracers of the degree of dissipation in spheroid formation.
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