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. V (Q) pQCD  $_{v}$ (M. Peter) CSR and  g1 Q (GeV) g1 (Q) $_{g1}$ from  $_{MS}$ + Bjorken sum rule $_{g1}$ from  $_{MS}$ + CSR $_{g1}$ from  $_{V}$ using pQCD (Peter)  $_{MS}$  g1 CSR CSR $_{g1}$ from  $_{V}$ using pQCD (Peter)  g1 CSR 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 0 0.25 0.5 0.75 1 1.25 1.5 1.75 1 2 3 4 5 6 7 8 9 10 FIG. 7. Check of the CSR validity. Top panel: Comparison of α$_{V}$ ( Q $^{2}$) from the two-loop pQCD calculation of Ref. [61] and α$_{V}$ ( Q $^{2}$) obtained using the CSR with α$_{g}$$_{1}$ as input. Bottom panel: Comparison of α$_{g}$$_{1}$ ( Q $^{2}$) computed using four different methods. The good agreement on top and bottom panels is a fundamental check of QCD. 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 24