The three-wave interaction is of strength 1 / α α (the same α α as in prior sections, above). The four-wave interaction is of strength 1 / α α 2 and the six-wave interaction is of strength 1 / α α 4 . Based on general principles from correlation of interactions (stemming from the BBGKY hierarchy) one expects the thermalization time to run as the square of the interaction. Thus, the original FPUT lattice (of size 16, 32 or 64) will eventually thermalize, on a time scale of order 1 / α α 8 : clearly, this becomes a very long time for weak interactions α α ≪ ≪ 1 ; meanwhile, the FPUT recurrence will appear to run unabated. This particular result holds for these particular lattice sizes; the resonant four-wave or six-wave interactions for different lattice sizes may or may not mix together modes (because the Brillouin zones are of a different size, and so the combinatorics of which wave-vectors can sum to zero is altered.) Generic procedures for obtaining canonical transformations that linearize away the bound modes remain a topic of active research.