Can the 126 GeV boson be a pseudoscalar?

We test the possibility that the newly-discovered 126 GeV boson is a pseudoscalar by examining the correlations among the loop-induced pseudoscalar decay branching fractions to $\gamma\gamma$, $ZZ^*$, $Z\gamma$, and $WW^*$ final states in a model-independent way. These four decays are controlled by only two effective operators, so that the rates in $Z\gamma$ and $WW^*$ are predicted now that the rates in $\gamma\gamma$ and $ZZ^*,Z\gamma^* \to 4 \ell$ have been measured. We find that the pseudoscalar possibility is disfavored but not conclusively excluded. Experimental exclusion of the $Z\gamma$ decay to well below $\sigma/\sigma_{\rm SM} \sim 170$ or conclusive observation of the $WW^*$ decay near the Standard Model rate would eliminate the pseudoscalar possibility. The $Z\gamma$ exclusion should be possible using existing data. The only loophole in our argument is the possibility that the $4\ell$ signal comes from pseudoscalar decays to a pair of new neutral gauge bosons with mass near the $Z$ pole.