Fock picture of bosonification in condensates

Question

I want to understand how bosonification in a condensate must be interpreted in the Fock states picture

Say i have uncoupled fermions in a set of states $E_1$, $E_2$ ... over the vacuum $E_0$. They occupy all the levels up to the Fermi energy.

Now, i introduce some coupling between them so, they become effectively boson pairs. Are the $E_i$ energy states out of the picture for these bosons? My random guess is that fermions would couple in states like

$$ | E_1 \times E_2 \rangle - | E_2 \times E_1 \rangle $$ $$ | E_1 \times E_2 \rangle + | E_2 \times E_1 \rangle $$ $$ | E_3 \times E_4 \rangle - | E_4 \times E_3 \rangle $$ $$ | E_3 \times E_4 \rangle + | E_4 \times E_3 \rangle $$

And so on, but this is the aspect i'm not sure and i want clarification

What are the states where the boson pairs are condensing into??