This paper presents a trace formula for an index over the spectrum of four-dimensional superconformal field theories on \(S^3 \times\) time. The index, denoted as \(\mathcal{I}^W\), receives contributions from states invariant under at least one supercharge and captures all information about protected short representations in 4-dimensional superconformal field theories. For the \(\mathcal{N} = 4\) theory, the index is a function of four continuous variables. The authors compute this index at weak coupling using gauge theory and at strong coupling by summing over the spectrum of free massless particles in \(AdS_5 \times S^5\), finding perfect agreement at large \(N\) and small charges. The index does not reproduce the entropy of supersymmetric black holes in \(AdS_5\), but this is not a contradiction as it differs qualitatively from the partition function over supersymmetric states of the \(\mathcal{N} = 4\) theory. The paper also evaluates and studies the partition function over the chiral ring in the \(\mathcal{N} = 4\) Yang-Mills theory.This paper presents a trace formula for an index over the spectrum of four-dimensional superconformal field theories on \(S^3 \times\) time. The index, denoted as \(\mathcal{I}^W\), receives contributions from states invariant under at least one supercharge and captures all information about protected short representations in 4-dimensional superconformal field theories. For the \(\mathcal{N} = 4\) theory, the index is a function of four continuous variables. The authors compute this index at weak coupling using gauge theory and at strong coupling by summing over the spectrum of free massless particles in \(AdS_5 \times S^5\), finding perfect agreement at large \(N\) and small charges. The index does not reproduce the entropy of supersymmetric black holes in \(AdS_5\), but this is not a contradiction as it differs qualitatively from the partition function over supersymmetric states of the \(\mathcal{N} = 4\) theory. The paper also evaluates and studies the partition function over the chiral ring in the \(\mathcal{N} = 4\) Yang-Mills theory.