Spin foam with topologically encoded tetrad on trivalent spin networks

We explore discrete approaches in LQG where all fields, the gravitational tetrad, and the matter and energy fields, are encoded implicitly in a graph instead of being additional data. Our graph should therefore be richer than a simple simplicial decomposition. It has to embed geometrical information and the standard model. We start from Lisi's model. We build a trivalent graph which is an F4 lattice of 48-valent supernodes, reduced as trivalent subgraphs, and topologically encoding data. We show it is a solution for EFE with no matter. We define bosons and half-fermions in two dual basis. They are encoded by bit exchange in supernodes, operated by Pachner 2-2 move, and rest state can be restored thanks to information redundancy. Despite its 4 dimensional nature, our graph is a trivalent spin network, and its history is a pentavalent spin foam.