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In this first of two papers, we explain in detail the simplest example of abroader set of relations between apparently very different theories. Ourexample relates $\mathfrak{su}(2)$ $\mathcal{N}=4$ super Yang-Mills (SYM) to atheory we call "$(3,2)$." This latter theory is an exactly marginal diagonal$SU(2)$ gauging of three $D_3(SU(2))$ Argyres-Douglas (AD) theories. We beginby observing that the Schur indices of these two theories are related by analgebraic transformation that is surprisingly reminiscent of indextransformations describing spontaneous symmetry breaking on the Higgs branch.However, this transformation breaks half the supersymmetry of the SYM theory aswell as its full $\mathcal{N}=2$ $SU(2)_F$ flavor symmetry. Moreover, it doesso in an interesting way when viewed through the lens of the corresponding 2Dvertex operator algebras (VOAs): affine currents of the small $\mathcal{N}=4$super-Virasoro algebra at $c=-9$ get mapped to the $\mathcal{A}(6)$ stresstensor and some of its conformal descendants, while the extra supersymmetrycurrents on the $\mathcal{N}=4$ side get mapped to higher-dimensional fermioniccurrents and their descendants on the $\mathcal{A}(6)$ side. We prove theserelations are facets of an exact graded vector space isomorphism (GVSI) betweenthese two VOAs. This GVSI respects the $U(1)_r$ charge of the parent 4Dtheories. We briefly sketch how more general $\mathfrak{su}(N)$ $\mathcal{N}=4$SYM theories are related to an infinite class of AD theories viageneralizations of our example. We conclude by showing that, in this class oftheories, the $\mathcal{A}(6)$ VOA saturates a new inequality on the number ofstrong generators.

$\mathcal{N}=4$ SYM, Argyres-Douglas Theories, and an Exact Graded Vector Space Isomorphism