The purpose of this paper is to explain the pattern of fill factors observed in the Fractional Quantum Hall Effect (FQHE), which appears to be restricted to odd-integer denominators as well as the sole even-integer denominator of 2. The method is to use the mathematics of gauge theory to develop Dirac monopoles without strings as originally taught by Wu and Yang, while accounting for orientation / entanglement and related “twistor” relationships between spinors and their environment in the physical space of spacetime. We find that the odd-integer denominators are included and the even-integer denominators are excluded if we regard two fermions as equivalent only if both their orientation and their entanglement are the same, i.e., only if they are separated by 4π not 2π. We also find that the even integer denominator of 2 is permitted because unit charges can pair into boson states which do not have the same entanglement considerations as fermions, and that all other even-integer denominators are excluded because only integer charges, and not fractional charges, can be so-paired. We conclude that the observed FQHE fill factor pattern can be fundamentally explained using nothing other than the mathematics of gauge theory in view of how orientation / entanglement / twist applies to fermions but not to bosons, while restricting all but unfractionalized fermions from pairing into bosons.