  This paper establishes and tests procedures which can determine the electron
energy gap of the high-temperature superconductors using the $t\!-\!J$ model
with spinon and holon quasiparticles obeying fractional statistics. A simpler
problem with similar physics, the spin susceptibility spectrum of the spin 1/2
fractional-statistics gas, is studied. Interactions with the density
oscillations of the system substantially decrease the spin gap to a value of
$(0.2 \pm 0.2)$ $\hbar \omega_c$, much less than the mean-field value of
$\hbar\omega_c$. The lower few Landau levels remain visible, though broadened
and shifted, in the spin susceptibility. As a check of the methods, the
single-particle Green's function of the non-interacting Bose gas viewed in the
fermionic representation, as computed by the same approximation scheme, agrees
well with the exact results. The same mechanism would reduce the gap of the
$t\!-\!J$ model without eliminating it.
