made of four vertices inside the 4 tetrahedrons connected with each other and the boundary faces. Now there are the 6 boundary faces with representations labeled $j_{1}...j_{6}$, plus 4 internal faces, $j_{7}...j_{10}$, of the dual, (see Figure \[triangduallevel1\]).

![The dual 2-complex for the triangulation level 1 with four tetrahedrons, highlighting the external faces.[]{data-label="triangduallevel1"}](triangduallevel1.png)

What happens is that even with the 6 boundary spin representations fixed, the classical $SU(2)$ fusion rules don’t give a constraint on the spin representations on these internal faces, and the amplitude can diverge $$W_{\triangle_{4}}(j_{1},...,j_{6})=\sum_{j_{f}}\prod_{f=1}^{10}(-1)^{j_{f}}d(j_{f})\prod_{v=1}^{4}(-1)^{j_{v}}\{6j\},\label{eq:transitionamplitude4tets}$$ where each $\{6j\}$ now has 3 boundary spin representations and 3 internal ones. These internal spins are not fixed by triangular inequalities. All the four internal spin form an internal tetrahedron where the spins can flow without restriction $-$ the tetrahedron with the vertices 1, 2, 3 and 4 in Figure \[triangduallevel1\]. From results of LQG, the spin quantum numbers correspond to eigenvalues of geometrical operators such as lengths and areas, and so there is a divergence when the scale of the geometry becomes large, a kind of infrared divergence. This is possible because in Riemannian geometry the space can be strongly curved. The 4 internal faces form what is called a 