  We explore the quantum algebraic formalism of the gauge origami system in
$\mathbb{C}^{4}$, where D2/D4/D6/D8-branes are present. We demonstrate that the
contour integral formulas have free field interpretations, leading to the
operator formalism of $qq$-characters associated with each D-brane. The
$qq$-characters of D2 and D4-branes correspond to screening charges and
generators of the affine quiver W-algebra, respectively. On the other hand, the
$qq$-characters of D6 and D8-branes represent novel types of $qq$-characters,
where monomial terms are characterized by plane partitions and solid
partitions. The composition of these $qq$-characters yields the instanton
partition functions of the gauge origami system, eventually establishing the
BPS/CFT correspondence.
  Additionally, we demonstrate that the fusion of $qq$-characters of D-branes
in lower dimensions results in higher-dimensional D-brane $qq$-characters. We
also investigate quadratic relations among these $qq$-characters. Furthermore,
we explore the relationship with the representations, $q$-characters, and the
Bethe ansatz equations of the quantum toroidal $\mathfrak{gl}_{1}$. This
connection provides insights into the Bethe/Gauge correspondence of the gauge
origami system from both gauge-theoretic and quantum-algebraic perspectives.
  We finally present conjectures regarding generalizations to general toric
Calabi-Yau four-folds. These generalizations imply the existence of an
extensive class of $qq$-characters, which we call BPS $qq$-characters. These
BPS $qq$-characters offer a new systematic approach to derive a broader range
of BPS/CFT correspondence and Bethe/Gauge correspondence.
