  Recent advancement of rainbow tensor models based on their superintegrability
(manifesting itself as the existence of an explicit expression for a generic
Gaussian correlator) has allowed us to bypass the long-standing problem seen as
the lack of eigenvalue/determinant representation needed to establish the
KP/Toda integrability. As the mandatory next step, we discuss in this paper how
to provide an adequate designation to each of the connected gauge-invariant
operators that form a double coset, which is required to cleverly formulate a
tree-algebra generalization of the Virasoro constraints. This problem goes
beyond the enumeration problem per se tied to the permutation group, forcing us
to introduce a few gauge fixing procedures to the coset. We point out that the
permutation-based labeling, which has proven to be relevant for the Gaussian
averages is, via interesting complexity, related to the one based on the
keystone trees, whose algebra will provide the tensor counterpart of the
Virasoro algebra for matrix models. Moreover, our simple analysis reveals the
existence of nontrivial kernels and co-kernels for the cut operation and for
the join operation respectively that prevent a straightforward construction of
the non-perturbative RG-complete partition function and the identification of
truly independent time variables. We demonstrate these problems by the simplest
non-trivial Aristotelian RGB model with one complex rank-3 tensor, studying its
ring of gauge-invariant operators, generated by the keystone triple with the
help of four operations: addition, multiplication, cut and join.
