  Despite the power of supersymmetry, finding exact closed-form expressions for
the protected operator spectra of interacting superconformal field theories
(SCFTs) is difficult. In this paper, we take a step towards a solution for the
"simplest" interacting 4D $\mathcal{N}=2$ SCFT: the minimal Argyres-Douglas
(MAD) theory. We present two results that go beyond the well-understood Coulomb
branch and Schur sectors. First, we find the exact closed-form spectrum of
multiplets containing operators that are chiral with respect to any
$\mathcal{N}=1\subset\mathcal{N}=2$ superconformal subalgebra. We argue that
this "full" chiral sector (FCS) is as simple as allowed by unitarity for a
theory with a Coulomb branch and that, up to a rescaling of $U(1)_r$ quantum
numbers and the vanishing of a finite number of states, the MAD FCS is
isospectral to the FCS of the free $\mathcal{N}=2$ Abelian gauge theory. In the
language of superconformal representation theory, this leaves only the spectrum
of the poorly understood $\bar{\mathcal{C}}_{R,r(j,\bar j)}$ multiplets to be
determined. Our second result sheds light on these observables: we find an
exact closed-form answer for the number of $\bar{\mathcal{C}}_{0,r(j,0)}$
multiplets, for any $r$ and $j$, in the MAD theory. We argue that this
sub-sector is also as simple as allowed by unitarity for a theory with a
Coulomb branch and that there is a natural map to the corresponding sector of
the free $\mathcal{N}=2$ Abelian gauge theory. These results motivate a
conjecture on the full local operator algebra of the MAD theory.
