details of the individual orbital contributions to the magnetism as a function of the hybridization between the conduction and localized electrons, which is beyond the scope of the theorem.

We begin by introducing a notation which simplifies the identification of the different sites/orbitals. Since we depleted all $f$-orbitals from one sublattice, then we have two different types of $d$-orbitals: $d_{0}$ and $d_{1}$. The former (latter) correspond to the $d$-orbitals without (with) hybridization with $f$-orbitals, as displayed in Fig. \[Fig:Lattice\_half\]. In addition, the unit cell for this geometry is composed by three sites, one of each type, with unit vectors $\mathbf{a}_{1} = a (1,1) $ and $\mathbf{a}_{2}
= a (1,-1) $. Here $a$ is the distance between nearest $d_{0}$ and $d_{1}$ sites. We will choose $a=0.5$ so that the distance between neighboring $d_{0}$ (or $d_{1}$) sites is 1. Finally, for technical reasons we performed our simulations in a $L \times L$ square geometry, with the primitive cell having twice the size of the unit cell, i.e. containing six sites, thus the number of unit cells being $N_{cells}=2 \times L^2$.

![(Color online) Local moments on the three types of sites as functions of hybridization at a low temperature.[]{data-label="Fig:local_mom"}](local_moments.pdf)

We begin by summarizing 