Pauli principle and coupling term in Weizsäcker formula

[crick]

Consider the pairing term in Weizsäcker formula. Here https://en.wikipedia.org/wiki/Semi-empirical_mass_formula#Pairing_term it is claimed that:

Due to the Pauli exclusion principle the nucleus would have a lower
energy if the number of protons with spin up were equal to the number
of protons with spin down.

I don't understand how Pauli exclusion principle should be the cause of this. This term comes from spin-spin interaction (or "coupling"), but I do not see the link with the fact that protons (or neutrons) with the same quantum numbers cannot occupy the same quantum state within a quantum system simultaneously

 

[mfb]

Without the Pauli exclusion principle all protons (and all neutrons) would simply occupy the lowest energy state and you wouldn't see any paring effect. Due to this principle every energy level has only two states (spin up and down) which get filled after each other. Every odd proton (neutron) has to start the next energy level.