  We compute the scaling dimensions of a family of fixed-charge operators at
the infrared fixed point of the $O(N)$ model featuring cubic interactions in
$d=6-\epsilon$ for arbitrary $N$ to leading and subleading order in the charge
but to all orders in the couplings. The results are used to analyze the
conjectured equivalence with the $O(N)$ model displaying quartic interactions
at its ultraviolet fixed point. This is performed by comparing the cubic model
scaling dimensions against the known large $N$ results for the quartic model
and demonstrating that they match. Our results reinforce the conjectured
equivalence and further provide novel information on the finite $N$ physics
stemming from the computations in the cubic model just below 6 dimensions.
