  The closure constraint is a central piece of the mathematics of loop quantum
gravity. It encodes the gauge invariance of the spin network states of quantum
geometry and provides them with a geometrical interpretation: each decorated
vertex of a spin network is dual to a quantized polyhedron in $\mathbb{R}^{3}$.
For instance, a 4-valent vertex is interpreted as a tetrahedron determined by
the four normal vectors of its faces. We develop a framework where the closure
constraint is re-interpreted as a Bianchi identity, with the normals defined as
holonomies around the polyhedron faces of a connection (constructed from the
spinning geometry interpretation of twisted geometries). This allows us to
define closure constraints for hyperbolic tetrahedra (living in the
3-hyperboloid of unit future-oriented spacelike vectors in $\mathbb{R}^{3,1}$)
in terms of normals living all in $SU(2)$ or in $SB(2,\mathbb{C})$. The latter
fits perfectly with the classical phase space developed for $q$-deformed loop
quantum gravity supposed to account for a non-vanishing cosmological constant
$\Lambda>0$. This is the first step towards interpreting $q$-deformed twisted
geometries as actual discrete hyperbolic triangulations.
