Given a finite lattice L that can be embedded in the recursively enumerable (r.e.) Turing degrees (RT, ≤T), we do not in general know how to characterize the degrees d ε RT below which L can be bounded. The important characterizations known are of the L7 and M3 lattices, where the lattices are bounded below d if and only if d contains sets of "fickleness" >ω and ≥ωω respectively. We work towards finding a lattice that characterizes the levels above ω2 , the first non-trivial level after ω. We introduced a lattice-theoretic property called "3-directness" to describe lattices that are no "wider" or "taller" than L7 and M3 . We exhaust the 3-direct lattices L, but they turn out to also characterize the >ω or ≥ωω levels, if L is not already embeddable below all non-zero r.e. degrees. We also considered upper semilattices (USLs) by removing the bottom meet(s) of some 3-direct lattices, but the removals did not change the levels characterized. This leads us to conjecture that a USL characterizes the same r.e. degrees as the lattice on which the USL is based. We discovered three 3-direct lattices besides M3 that also characterize the ≥ωω -levels. Our search for a >ω2-candidate therefore involves the lattice-theoretic problem of finding lattices that do not contain any of the four ≥ωω-lattices as sublattices.