Extended Data Fig. 5: Stability and convergence of the chiral–continuum extrapolation.

In the left panel, the model-averaged result (‘model avg’) is the black square. The vertical magenta band is the resulting 68% confidence band. The next six values are results from individual extrapolations that go into the model average, described in Supplementary Information, section S.7A. Uncertainties are one s.e.m. ‘ct’, counter-term; ‘FV’, finite volume; ‘disc.’, discretization; α_S = g²/(4π), where g is the quark–gluon coupling of QCD. The middle panel shows the augmented χ² (\({\chi }_{{\rm{a}}{\rm{u}}{\rm{g}}}^{2}\)) per degree of freedom (dof), where \({\chi }_{{\rm{a}}{\rm{u}}{\rm{g}}}^{2}\) is the sum of the χ² values from the data and from the priors. All fits have 16 degrees of freedom because each prior is counted as a data point. The right panel shows the resulting Bayes factors normalized by the NLO Taylor \({\varepsilon }_{{\rm{\pi }}}^{2}\) Bayes factor, which is found to be the largest among them. These normalized Bayes factors are used as relative weights in the model-averaging procedure. The stability of the extrapolation analysis is tested by including additional discretization terms, omitting the predicted NLO finite-volume corrections, increasing the prior widths on the leading order (LO) and all low-energy constants, and applying cuts on the pion masses considered and on the discretization scales included. All variations are contained within 1σ of the model-average value, with most being substantially smaller than 1σ from the central value. Finally, we show the resulting extrapolation from the complete next-to-next-to-next-to-leading order (N3LO) chiral perturbation theory analysis and from the NLO chiral perturbation theory analysis with ∆ degrees of freedom (χPT(∆)). The N3LO fit is not included in the average because it has five unknown low-energy constants and we have only five different pion mass values. The NLO χPT(∆) value is not included because it requires input from phenomenology and is thus not a pure lattice QCD prediction, and also the next-to-next-to-leading order (NNLO) χPT(∆) extrapolation function is not known, so a test of stability and convergence is not possible.