  The continuum of holographic dual gravitational charges is recovered out of
the discrete spectrum of $U(N)$ $\mathcal{N}=4$ SYM on $\mathbb{R}\times
S^3\,$. 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. The computed mass-gap matches the conjectured strong-coupling
result obtained by Boruch, Heydeman, Iliesiu and Turiaci in supergravity. The
emergent reparameterizations, broken by 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 the way to
understanding the emergence of chaos in gauge theories and its relation to
non-extremal and non-supersymmetric black hole physics.
