  Black holes are chaotic quantum systems that are expected to exhibit random
matrix statistics in their finite energy spectrum. Lin, Maldacena, Rozenberg
and Shan (LMRS) have proposed a related characterization of chaos for the
ground states of BPS black holes with finite area horizons. On a separate
front, the "fuzzball program" has uncovered large families of horizon-free
geometries that account for the entropy of holographic BPS systems, but only in
situations with sufficient supersymmetry to exclude finite area horizons. The
highly structured, non-random nature of these solutions seems in tension with
strong chaos. We verify this intuition by performing analytic and numerical
calculations of the LMRS diagnostic in the corresponding boundary quantum
system. In particular we examine the 1/2 and 1/4-BPS sectors of $\mathcal{N}=4$
SYM, and the two charge sector of the D1-D5 CFT. We find evidence that these
systems are only weakly chaotic, with a Thouless time determining the onset of
chaos that grows as a power of $N$. In contrast, finite horizon area BPS black
holes should be strongly chaotic, with a Thouless time of order one. In this
case, finite energy chaotic states become BPS as $N$ is decreased through the
recently discovered "fortuity" mechanism. Hence they can plausibly retain their
strongly chaotic character.
