  The energy gaps appearing in the fractional quantum Hall effect (FQHE) remain
an essential aspect of the investigation. Moreover, the plateau widths in the
Hall resistance have been considered simply an effect of disorder as in the
integral quantum Hall effect. The existing theories could neither explain the
Hall resistance curve owing to plateau widths nor calculate the energy gaps.
This study reveals that both the energy gaps and plateau widths contain
fundamental many-body aspects of the FQHE. They are found to be connected via
the strengths of multi-particle correlations, which do not affect the plateau
heights. They are automatically quantized just by the presence of
multi-particle correlations. This work focuses on correlated skipping electrons
moving through the edge of an incompressible strip formed within a Hall bar.
Consequently, a single-particle Hamiltonian was constructed incorporating the
Zeeman energies of multiply-correlated skipping electrons. The resulting energy
spectrum exhibits hierarchical splits of the Landau levels according to
correlation order. The lowest Landau level is examined. Based on such level
splitting, a previously measured Hall resistance curve and energy gaps are
quantitatively explained by determining the parameters that describe the
degrees of multi-particle correlations. The chemical potential and effective
$g$-factors are additionally predicted for the Hall resistance. Furthermore,
the fractional electron charge $e/(2n+1)$ for an electron participating in
$n$-particle correlation was obtained by identifying the Fermi distribution
function of $n$ correlated basic transport entities moving through the edge of
the incompressible strip. Finally, the ideal-like Hall resistance was obtained
at half-filling using the strengths of multi-particle correlations given in a
regular pattern.
