According to heuristic arguments, global AdS$\mathrm{<mrow\; data-mjx-texclass="ORD">5</mrow>}$×S$\mathrm{<mrow\; data-mjx-texclass="ORD">5</mrow>}$ black holes are expected to undergo a phase transition in the microcanonical ensemble. At high energies, one expects black holes that respect the symmetries of the S$\mathrm{<mrow\; data-mjx-texclass="ORD">5</mrow>}$; at low energies, one expects "localized" black holes that appear pointlike on the S$\mathrm{<mrow\; data-mjx-texclass="ORD">5</mrow>}$. According to anti-de Sitter/conformal field theory correspondence, N=4 supersymmetric Yang-Mills (SYM) theory on a 3-sphere should therefore exhibit spontaneous R-symmetry breaking at strong coupling. In this Letter, we numerically construct these localized black holes. We extrapolate the location of this phase transition, and compute the expectation value of the broken scalar operator with lowest conformal dimension. Via the correspondence, these results offer quantitative predictions for N=4 SYM theory.