  We derive the trial Hall resistance formula for the quantum Hall metals to
address both the integer and fractional quantum Hall effects. Within the
degenerate Landau levels, Zeeman splitting and level crossings in the presence
of changing magnetic-field strength determine the Pancharatnam phase
retardation, including the phase acceleration or deceleration, which are
related to the changes in the phase and group momenta of a wavefunction. We
discuss the relevant physical postulates with respect to Pancharatnam phase
retardation to qualitatively reproduce the measured Hall resistance's zigzag
curve for both the integer and the fractional filling factors. Along the way,
we give out some hints to falsify our postulates with experiments.
