  Canonical analysis of a recently proposed [1] linear+quadratic curvature
gravity model in D=3 displays its pure fourth derivative quadratic branch as a
ghost-free (massless) excitation. Hence it both negates an old no-go theorem
and is power-counting UV finite. It is also conformal-invariant, so the metric
is underdetermined. While the 2-term branch is also ghost-free, it has, as
shown in [1], a second-derivative, two-tensor equivalent, akin to the second
order scalar-tensor form of ostensibly fourth order, $R+R^2$, actions. This
correspondence fails for the pure quadratic branch: it is irreducibly
fourth-order.
