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Degeneracy means the state of being abnormal, generally carrying one of the implications, "normal rules do not apply" or "the situation is ambiguous". It is conventionally used regarding a number of cases within physics and astrophysics that aren't otherwise closely related, and use of the same word for these different cases has some potential to confuse readers.
The term is used regarding degenerate matter such as electron degenerate matter or neutron degenerate matter, in which degeneracy can be taken to indicate the equation of state is not the "normal" ideal gas law. In this case, degeneracy has also taken on the meaning of the quantum circumstance that underlies this divergence. Electron degenerate matter's resistance to compression is termed electron degeneracy pressure.
The phrase quantum degeneracy can indicate quantum states having the same (or very close) energy levels (degenerate energy levels), thus violating the normal idea that at a given energy level, the quantum state is determined. Electron orbits may exhibit degenerate energy levels, i.e., two different quantum states at the same energy level, exhibiting the same spectral lines or spectral lines that are very close together, or a low frequency spectral line from when one changes to the other such as hydrogen's 21-cm line.
The term degeneracy is also used to indicate an ambiguity in analysis, i.e., an analysis that produces multiple values, most (e.g., all but one) being incorrect. As in the statement: "There is a degeneracy in kinematic distance determination." Phrases regarding breaking a degeneracy describe some type of observation or analysis that resolves an ambiguity, e.g., reveals which of two conflicting conclusions is the correct one.