How is Pauli's exclusion principle conserved in transformations from real space to $k$ space for a Fock space hopping Hamiltonian?

$$hat{H}= -tsum_{langle i,jrangle} c_{isigma}^{+}c_{jsigma}+h.c$$

For this Hamiltonian a lattice with all sites doubly occupied would give 0 in real space and for single occupancy it gives $-4t^{2}L$ following Pauli’s exclusion which has been built into the Fock space operator however on doing a Fourier transform on the term we get

$$ -2t sum_{k} (cos(kx)+cos(ky)) c_{ksigma}^{+}c_{ksigma}$$
However I do not see how this shows the effect of Pauli’s exclusion and for both an infinite lattice with doubly occupied sites and singly occupied sites this will give the same value. I might have misunderstood something but I do not understand how both their values are different for an infinite lattice. Shouldn’t there be an additional multiplication factor coming in this k-space term. In that case what should it be?

This is on a Hubbard like model for a system which does not have the interaction term.