  Building upon previous $2D$ studies, this research focuses on describing $3D$
tensor renormalisation group (RG) flows for lattice spin systems, such as the
Ising model. We present a novel RG map, which operates on tensors with
infinite-dimensional legs and does not involve truncations, in contrast to
numerical tensor RG maps. To construct this map, we developed new techniques
for analysing tensor networks. Our analysis shows that the constructed RG map
contracts the region around the tensor $A_*$, corresponding to the
high-temperature phase of the $3D$ Ising model. This leads to the iterated RG
map convergence in the Hilbert-Schmidt norm to $A_*$ when initialised in the
vicinity of $A_*$. This work provides the first steps towards the rigorous
understanding of tensor RG maps in $3D$.
