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## Appendix: Consistency Check of Commensurate Scale Relations 

In this appendix we verify, within the uncertainties discussed in the text, the validity of the CSR predictions in the pQCD domain. 

<img src='content_image/346882.jpg'>

The verification of CSR for different schemes is illustrated on Fig. 7. On the top panel of Fig. 7 we compare the full two-loop computation of α$_{V}$ ( Q $^{2}$) from Ref. [61] with the coupling α$_{V}$ ( Q $^{2}$) resulting from applying the CSR to α$_{g}$$_{1}$ down to Q 2 = 0 . 6 GeV $^{2}$. The width of the bands gives the uncertainties. For the sparse cross-hatched band (two-loop pQCD calculation), the uncertainty stems from Λ$_{QCD}$, the truncation of the α$_{MS}$ series to β$_{2}$ , and the truncation of the α$_{V}$ series to two loops ( a$_{2}$ coefficient) in [61]. All these contributions are added in quadrature. For the dense cross-hatched band (CSR), the uncertainties come from Λ$_{QCD}$, the truncation of the α$_{MS}$ series to β$_{2}$ , and the truncation of the CSR series to 

