  The microstructure of black holes is a mystery. There is yet no resolution of
basic questions such as what the constituent particles are. We work here with
black hole thermodynamics (BHT), and the metric geometry of thermodynamics,
which connects to interparticle interactions via the invariant thermodynamic
Ricci scalar curvature $R$. $R$ may be calculated with BHT. In ordinary
thermodynamics (OT), $R$ is positive/negative for interparticle interactions
repulsive/attractive. Its magnitude is the correlation length. The basic
universality of thermodynamics leads us to expect similar relations for BHT.
Our contribution here is motivated by a physical simplification that frequently
occurs at low temperatures $T$ in OT: complicated interactions tend to freeze
out, leaving only basic quantum statistical interactions, those of ideal Fermi
or Bose gasses. Our hope is that similar simplification happens in black holes
in the extremal limit, where the BHT temperature $T \to 0$. We evaluate the
extremal $R$ for twelve BHT literature models, working with the independent
variables mass, angular momentum, charge, and the cosmological constant,
$\{M,J,Q,\Lambda\}$, respectively. We allowed only two of these variables to
fluctuate at a time, with the other two fixed. $M$ always fluctuated, either
$J$ or $Q$ fluctuated, and $\Lambda$ was always fixed. At constant average $M$,
$R$ has limiting divergence $R=c\,T^{-1}$, with the nonsingular constant $c$
depending only on $M$ and the two fixed parameters. $c$ is positive for $11/12$
of the models we examined, and negative only for the tidal charged model. The
positive sign for $R$ indicates a BHT microstructure composed of particles with
repulsive (fermionic) interactions. The limiting BHT expression for $R$
resembles that for 2D and 3D ideal Fermi gasses at constant volume, which also
have limiting divergence $R=c\,T^{-1}$, with positive $c$.
