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Electron degeneracy is a quality of matter occurring when it is compressed to the point that electrons fill the lowest quantum states, forming electron degenerate matter (EDM). The Pauli exclusion principle dictates that within a sufficiently small volume, at most one electron can be at any specific state. The result is a pressure (electron degeneracy pressure aka degenerate electron pressure) against further compression. Another way to look at it (that explains more to me) is that the Heisenberg uncertainty principle indicates that beyond some point, any further confinement of electrons can occur only if their momentum is increased, i.e., their speed must rise, which requires energy, a requirement manifested as a resistance to compression that must be overcome: the energy to make progress against this resistance provides the energy for accelerating the electrons, thus the pressure. During compression, as electrons take on momentum, they gain too much to maintain a nucleus orbit, and they travel freely through the material rather than remain with a single nucleus.
The word degeneracy is used to indicate that at this compression, the ideal gas law no longer holds: specifically, the temperature no longer has the same relation to pressure and volume. The pressure provides the force that keeps a white dwarf (and the core of some giant stars and gas giants) from collapsing further, e.g., into a neutron star. An analogous kind of degenerate matter and degeneracy pressure is neutron degenerate matter and neutron degeneracy pressure, for analogous neutron quantum states, which describes the material of neutron stars and possibly the center of some white dwarfs. With sufficient gravity, these pressures can be overcome, producing a black hole. Though it isn't clear what the microscopic conditions are in the extreme conditions underneath the event horizon and/or approaching the singularity, it is assumed the matter is forcibly transformed to bosons (e.g., photons), which are not ruled by the Pauli exclusion principle.
Given electron degeneracy, the Fermi energy is the energy level of the highest quantum state occupied if the material is at absolute zero.