Our particular synthesis of this result is that a general large spin transition amplitude should be built from “gluing” quantum building blocks $W_{\triangle_{1}}$ given by $\{6j\}_{q}$ with a specific $q$ that fixes $W_{\triangle_{1}}$ to low spin representations, and we will justify in the following the choice $q=e^{\frac{2\pi}{5}i}$.

The Quantum Tetrahedron with Topological Symmetry {#qt}
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We propose to derive the large spin transition amplitude previously discussed from the direct quantization of geometry, in particular a quantum building block, a tetrahedron, following the notion of quantum tetrahedrons [@Barbieri]. The quantization of one tetrahedron allows us to define and address quantization of geometric quantities like lengths, areas and volumes. Yet the usual quantization with $SU(2)$ symmetry is much too general, allowing an infinity of values for these geometric quantities. We therefore follow the conceptual discussion in [@KleeCodetheoreticaxiom], which states that the Planck scale regime is a code $-$ *(1) a finite set of symbolic objects, with (2) ordering rules and (3) syntactical freedom (4) for the purpose of expressing meaning, e.g., self-referential physical observables and expected values*. More specifically, the syntactical rules of the code are related to restrictions on the representation theory of Lie algebras [@5throot], which can be implemented by 