  The duality between type IIA superstring theory and M-theory enables us to
lift bound states of D$0$-branes and $n$ parallel D$6$-branes to M-theory
compactified on an $n$-centered multi-Taub-NUT space $\mathbb{TN}_{n}$.
Accordingly, the rank $n$ K-theoretic Donaldson-Thomas invariants of
$\mathbb{C}^{3}$ are connected with the index of M-theory on
$\mathbb{C}^{3}\times\mathbb{TN}_{n}$. In this paper, we extend this connection
by considering intersecting D$6$-branes. In the presence of a suitable
Neveu-Schwarz $B$-field, the system preserves two supercharges. This system is
T-dual to the configuration of tetrahedron instantons which we introduced in
\cite{Pomoni:2021hkn}. We conjecture a closed-form expression for the
K-theoretic tetrahedron instanton partition function, which is the generating
function of the D$0$-D$6$ partition functions. We find that the tetrahedron
instanton partition function coincides with the partition function of the
magnificent four model for special values of the parameters, leading us to
conjecture that our system of intersecting D$6$-branes can be obtained from the
annihilation of D$8$-branes and anti-D$8$-branes. Remarkably, the K-theoretic
tetrahedron instanton partition function allows an interpretation in terms of
the index of M-theory on a noncompact Calabi-Yau fivefold which is related to a
superposition of Kaluza-Klein monopoles. The dimensional reduction of the
system allows us to express the cohomological tetrahedron instanton partition
function in terms of the MacMahon function, generalizing the correspondence
between Gromov-Witten invariants and Donaldson-Thomas invariants for Calabi-Yau
threefolds.
