  We present an exact RG (renormalization group) analysis of $O(N)$-invariant
scalar field theory about the Gaussian fixed point. We prove a series of
statements that taken together show that the non-polynomial eigen-perturbations
found in the LPA (local potential approximation) at the linearised level, do
not lead to new interactions, \textit{i.e.} enlarge the universality class,
neither in the LPA or treated exactly. Non-perturbatively, their RG flow does
not emanate from the fixed point. For the equivalent Wilsonian effective action
they can be re-expressed in terms of the usual couplings to polynomial
interactions, which can furthermore be tuned to be as small as desired for all
finite RG time. For the infrared cutoff Legendre effective action, this can
also be done for the infrared evolution. We explain why this is nevertheless
consistent with the fact that the large field behaviour is fixed by these
perturbations.
