The purpose of this paper is to explain the pattern of fill factors observed in the Fractional Quantum Hall Effect (FQHE) 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 fully develop Dirac monopoles without strings as originally taught by Wu and Yang, while accounting for topological 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 permitted and the even-integer denominators excluded if FQHE only displays electrons of identical orientation-entanglement “version,” i.e., only electrons separated by 4π not 2π. We also find that the even-integer denominator of 2 is permitted if entangled electrons can pair into boson states, and that all other even-integer denominators are excluded because bosons are not subject to the same Exclusion statistics as are fermions. Because this proposed relation between the Dirac monopoles and the FQHE presupposes an electric / magnetic duality near 0K, and because magnetic monopoles are certainly not observed at higher temperatures, we also find how to break this duality symmetry with the consequence that the low-temperature Dirac monopoles are replaced by a “thermal residue” at higher temperatures. 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 orientation, entanglement and twist, with proper breaking of the low-temperature electric / magnetic duality. An unanticipated bonus is that the quantum topology emerging from this analysis appears to map precisely to the electronic orbital structure of atoms. This provides the basis for proposed experiments to closely observe the FQHE quasiparticles to seek correlations to the angular momentum observed in atomic electron shells, and to boson spin states.