The continuum of holographic dual gravitational charges is recovered out of the discrete spectrum of U(N) \( \mathcal{N} \) = 4 SYM on ℝ × S3. In such a limit, the free energy of the free gauge theory is computed up to logarithmic contributions and exponentially suppressed contributions. Assuming the supergravity dual prediction to correctly capture strong-coupling results in field theory, the answer is bound to encode a complete low-temperature expansion of the Gibbons-Hawking gravitational on-shell action, valid well beyond the vicinity of supersymmetric black hole solutions. The formula recovers the long awaited Schwarzian contribution at low enough temperatures for certain choices of flows to the continuum. One such choice identifies the chemical potentials and thermodynamic charges in field theory with the chemical potentials and thermodynamic charges of the dual black holes. For such a flow the computed mass-gap kinematically matches the conjectured strong-coupling result obtained by Boruch, Heydeman, Iliesiu, and Turiaci in supergravity, including small 1/λ-corrections. The emergent reparameterizations, broken by the selection of the Schwarzian, correspond to redefinitions of the relevant cutoff scale. Observations are made regarding the existence of \( \frac{1}{8} \) -BPS black holes and how this is in tension with BPS inequalities. The RG-flow procedure leading to these results opens a way to understanding the emergence of chaos in gauge theories and its relation to non-extremal and non-supersymmetric black hole physics.
