= Odd filling obstructs a spin-degenerate band insulator
In an independent-electron crystal with two spin states and no spin splitting, each completely filled <energy band> contains two <electrons> per primitive cell. An odd <electron> count per primitive cell therefore cannot fill an integer number of such bands. Enlarging the actual primitive cell, breaking the relevant degeneracy or including correlated-electron physics changes the premises. Even filling is necessary in this model but not sufficient: bands can overlap.
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