We make ab initio predictions for the A = 6 nuclear level scheme based on two- and three-nucleon interactions up to next-to-next-to-leading order in chiral effective field theory ($chi$EFT). We utilize eigenvector continuation and Bayesian methods to quantify uncertainties stemming from the many-body method, the $chi$EFT truncation, and the low-energy constants of the nuclear interaction. The construction and validation of emulators is made possible via the development of JupiterNCSM -- a new M-scheme no-core shell model code that uses on-the-fly Hamiltonian matrix construction for efficient, single-node computations up to $N_mathrm{max} = 10$ for ${}^{6}mathrm{Li}$. We find a slight underbinding of ${}^{6}mathrm{He}$ and ${}^{6}mathrm{Li}$, although consistent with experimental data given our theoretical error bars. As a result of incorporating a correlated $chi$EFT-truncation errors we find more precise predictions (smaller error bars) for separation energies: $S_d({}^{6}mathrm{Li}) = 0.89 pm 0.44$ MeV, $S_{2n}({}^{6}mathrm{He}) = 0.20 pm 0.60$ MeV, and for the beta decay Q-value: $Q_{beta^-}({}^{6}mathrm{He}) = 3.71 pm 0.65$ MeV. We conclude that our error bars can potentially be reduced further by extending the model space used by JupiterNCSM.
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