This question is specifically about Schrödinger quantum mechanics, but if an answer in some other mode would illuminate it could be acceptable, as demonstrating a physical or mathematical reason for added axioms.
In short - since the p-orbital has rotational symmetry about only one axis, but the potential of a point charge has spherical symmetry, a specific solution corresponding to a p-orbital should also be a solution when arbitrarily rotated. That means there is an infinite number of p-orbital solutions in this context. However, the dimension of the solution space for the given energy, that is, the eigenspace for the given eigenvalue is presumably exactly three. One can use three axial p-orbitals to span the whole eigenspace.
Thus the exclusion principle for fermions seems to be that there are at most the dimension of the eigenspace number of particles in an eigenspace rather than 1 particle in an orbital, if an orbital is taken as a solution to the Schrödinger equation.
Can anyone confirm or deny this line of reasoning? And provide a reference to explicit statement in the literature?