  We place unprecedented constraints on recoil corrections in the $\beta$ decay
of $^8$Li, by identifying a strong correlation between them and the $^8$Li
ground state quadrupole moment in large-scale ab initio calculations. The
results are essential for improving the sensitivity of high-precision
experiments that probe the weak interaction theory and test physics beyond the
Standard Model (BSM). In addition, our calculations predict a $2^+$ state of
the $\alpha+\alpha$ system that is energetically accessible to $\beta$ decay
but has not been observed in the experimental $^8$Be energy spectrum, and has
an important effect on the recoil corrections and $\beta$ decay for the $A=8$
systems. This state and an associated $0^+$ state are notoriously difficult to
model due to their cluster structure and collective correlations, but become
feasible for calculations in the ab initio symmetry-adapted no-core shell-model
framework.
