We know that the non-relativistic system like Graphene with a particular filling of electrons give rise to relativistic Dirac equations at low energy, with multi-flavors of Dirac particles.
In terms of free theory specified by the $k$-spaces, the different flavors of Dirac particles may live on the different points (say $K,K'$ points) in the $k$-space Brillouin zone.
At the low energy, we may live in the vacuum with several flavors of Dirac particles (quarks, leptons, etc) similar to the one in a certain non-relativistic system like Graphene (but we are in the 3+1 dimensional Dirac semi-metal instead of 2+1 dimensional Graphene).
Let us call a non-relativistic Dirac semi-metal-like/Graphene-like system $A$, and the relativistic system with Dirac particles as the system $B$.
My question is that can we tell apart $A$ from $B$? If God or some physicists told us that we actually live in a relativistic system $B$, could we determine at low energy that we are living in a non-relativistic system $A$ where the relativistic phenomenon of $B$ is just a low-energy emergent phenomenon?
And the other way around, suppose some physicists told us that the ground state of non-relativistic Dirac semi-metal-like/Graphene-like system (for instance at half-filling of electrons) are the same and can-not be distinguished from the relativistic systems of Dirac fermions in QFT. Could we justify or falsify the statements? For example, the two-flavor free Dirac fermions in relativistic QFT (system $A$) in the sub-atomic particle physics, do not require the labels of special $K,K'$ points in the $k$-space Brillouin zone for the system $B$. So do those special points imply some subtle difference between the systems $A$ and $B$?