In this chapter we treat the remaining deep analytical results which are needed for a robust viscosity theory for subharmonics in general potential theories and subsolutions of fully nonlinear elliptic PDEs. We address the central problem of how to ensure a priori the existence of a sufficient amount of upper contact points for locally semiconvex functions which, when combined with Alexandrov’s Theorem and the sup-convolution approximation of semicontinuous functions by semiconvex functions, will complete the foundations of the “hard analysis” part of the viscosity theory. The analysis will culminate in the so-called Jensen–Słodkowski Theorem 3.27, which unifies and generalizes two historically distinct approaches (Jensen’s Lemma and Słodkowski’s density estimate) to the central problem in a viscosity approach to potential theory and operator theory. Several important consequences of the Jensen–Słodkowski Theorem are also given, as well as a novel proof of this fundamental theorem which is shown to follow from the area formula for gradients of locally semiconvex functions.