# Birotons and \"Dark\" Quantum Hall Hierarchies
## 5 Conclusions  
In the conclusion of the study, we constructed the hierarchy of computable fractional QHE states in the range of electron filling factors from \(1/4\) to \(3/4\) (Figure 4). The energies of these fractional states were found to be reasonably consistent with the Zang-Birman hierarchical structure--Farey-number hierarchy structure [12]. Although this particular structure has much in common with the theoretical models of [PERSON], [PERSON], and [PERSON], it has a significant distinctive feature. Fractional states of highly varied denominator values from different FQHE hierarchies can have nearly equal energy gaps between the ground and excited states. For instance, the energies of such dissimilar fractional states as \(6/19\), \(6/17\), \(8/23\), \(8/25\), \(10/29\), and \(10/31\) belonging to different \"dark\" hierarchies are found to be almost the same, as all of them belong to the second steps from the top of their hierarchical ladders--the Laughlin state \(1/3\) (Figure 4). Conversely, the energies of fractional QHE states with equal denominators and similar electron filling factors can differ substantially. For example, the respective biroton energies of fractional states \(10/31\) and \(11/31\), corresponding to the second and third steps from the top of their hierarchical ladders, come to be nearly two-orders of magnitude apart (\(51.8\cdot 10^{-3}\) and \(0.74\cdot 10^{-3}\) in Coulomb units (Figure 3), respectively). Despite the abundance of calculated fractional QHE states from \"dark\" hierarchies, the fact that they are not directly observable in magneto-transport experiments is quite understandable. The hierarchical structure proposed in the present work makes it evident that biproton energies of fractional states in the Laughlin-Jain hierarchy are always greater than other fractional states of comparable filling factors belonging to \"dark\" hierarchies. Hence, it is more energy favorable to localize some part of excited quasi-electrons and quasi-holes of Laughlin-Jain fractional states and to keep the filling factor of extended states unchanged in the region of filling factors separating the neighboring fractional states of this hierarchy. Considering that even a small amount of impurities in a two-dimensional system causes localization of a macroscopic number of electron states, the filling factor regions where we can observe the fractional states of \"dark\" hierarchies are not large, even in the most highly mobile samples known to this day [28]. Usually, for \(\
u<1\), these regions fall within narrow ranges of electron filling factors, [\(1/3\); \(2/5\)] and [\(3/5\); \(2/3\)], where the Laughlin-Jain fractional states are far apart in terms of the filling factor. As the electron filling factor approaches \(1/2\), free of Laughlin-Jain fractional states, regions shrink in size because the \"separation\" (in terms of the electron filling factor) between the nearby fractional states of the Laughlin-Jain hierarchy decreases as the inverse square of the fraction's denominator. Therefore, while the Laughlin-Jain hierarchy is not the only hierarchy of FQHE states, it is still dominant over the \"dark\" hierarchies in magnetotransport experiments (Figure 4). The \"dark\" hierarchies become, in turn, essential in describing the excitation properties of FQHE states. The \"dark\" hierarchies are nevertheless essential in describing the excitation properties of FQHE states. Taking into account transitions between the FQHE states of the Laughlin-Jain hierarchy and the states of the \"dark\" hierarchies, the non-physical negative values for activation energies close to the filling factor \(1/2\)[7] are explained. The abnormal excitation energies measured by resonant Raman scattering for the fractional states, \(2/5\) and \(3/7\)[6], as well as the spin excitation energies for the state \(1/3\) measured by the authors of this paper [5; 26; 27] are similarly explained,.