  Recent anomalies in $^8$Be and $^4$He nuclear decays can be explained by
postulating a fifth force mediated by a new boson $X$. The distributions of
both transitions are consistent with the same $X$ mass, 17 MeV, providing
kinematic evidence for a single new particle explanation. In this work, we
examine whether the new results also provide dynamical evidence for a new
particle explanation, that is, whether the observed decay rates of both
anomalies can be described by a single hypothesis for the $X$ boson's
interactions. We consider the observed $^8$Be and $^4$He excited nuclei, as
well as a $^{12}$C excited nucleus; together these span the possible $J^P$
quantum numbers up to spin 1. For each transition, we determine whether scalar,
pseudoscalar, vector, or axial vector $X$ particles can mediate the decay, and
we construct the leading operators in a nuclear physics effective field theory
that describes them. Assuming parity conservation, the scalar case is excluded
and the pseudoscalar case is highly disfavored. Remarkably, however, the
protophobic vector gauge boson, first proposed to explain only the $^8$Be
anomaly, also explains the $^4$He anomaly within experimental uncertainties. We
predict signal rates for other closely related nuclear measurements, which, if
confirmed, will provide overwhelming evidence that a fifth force has been
discovered.
