  Compact lattice Quantum Electrodynamics (QED) with four species of fermions
is simulated with massless quarks by using the $\chi$QED scheme of adding a
four-fermi interaction to the action. Simulations directly in the chiral limit
of massless quarks are done with high statistics on $8^4$, and $16^4$ lattices,
and the phase diagram, parameterized by the gauge and the four-fermi couplings,
is mapped out. The line of monopole condensation transitions is separate from
the line of chiral symmetry restoration. The simulation results indicate that
the monopole condensation transition is first order while the chiral transition
is second order. The challenges in determining the Universality class of the
chiral transition are discussed. If the scaling region for the chiral
transition is sufficiently wide, the $16^4$ simulations predict critical
indices far from mean field values. We discuss a speculative scenario in which
anti-screening provided by double-helix strands of monopole and anti-monopole
loops are the agent that balances the screening of fermion anti-fermion pairs
to produce an ultra-violet fixed point in the electric coupling.
