A classical Pauli exclusion principle for electrons

Let us consider the following model. We have a 2-dimensional plate with a uniform positive electric charge density. We put electrons close to the plate, so that the positive and negative electric charges cancel out.

                          electrons

             ● e-     ● e-   ● e- ● e- ● e-

          +          +          +          +          +

                               plate

What happens if we let the electrons have zero or very little kinetic energy?

If the kinetic energy is zero, then the electrons will form a "crystal" where they minimize their repulsive potential energy. We could say that each of the electrons is in a different "state" from others, since the electrons are static and at different positions. A Pauli exclusion principle holds in this case. The electron system is in a "solid" state.

If we then add a little bit of kinetic energy, the electrons probably will vibrate around their initial positions. We can still say that that the electrons have different states.

Adding still more energy, the electrons can slide past each other. The electron system is in a "liquid" state. The electrons are in different states, but can exchange their states easily.

If we add still more energy, we will have an electron gas where the electrons bounce around. A gas has a certain distribution D of energies. We could say that the distribution D is the "Fermi sea" of the classical electrons.

Can we squeeze a very large number of electrons on the plate, such that the energy distribution D stays unchanged?

Probably not. The repulsive force between the electrons grows stronger as we add more electrons. The system will become a liquid or a solid again.

We showed that there may exist a classical Pauli exclusion principle: the distribution D can only fit a certain number of electrons before the electrons start to crystallize into separate (almost) local states. As if the distribution D would only contain a finite number of "separate states" for electrons.

We have a classical analogue of the quantum mechanical Pauli exclusion principle. The classical principle follows from the simple fact that electrons repel each other.

https://en.wikipedia.org/wiki/Spin%E2%80%93statistics_theorem

The Pauli exclusion principle in quantum mechanics is a consequence of the spin-statistics theorem. The dubious "proof" of the theorem in Wikipedia uses the algebra of spin rotations. A non-integer spin implies the Pauli exclusion principle while an integer spin allows many particles to have the same state.

https://en.wikipedia.org/wiki/Cooper_pair

In a superconductor, electrons are believed to form bound Cooper pairs. A pair has an integer spin, and many pairs can exist in the same "state". 

What is the relation of these two things: having a non-integer spin and having a repulsive or attractive force between particles? We need to investigate this.