to consistently extend the AdS coupling to the UV domain, consistent with pQCD. VIII. HOLOGRAPHIC COUPLING IN CONFIGURATION SPACE A. Comparison of V and g$_{1}$ Results In order to obtain modifications to the instantaneous Coulomb potential in configuration space V ( r ) = − C$_{F}$ α$_{V}$ ( r ) /r from the running coupling, one must transform the coupling defined by the static quark potential V ( q ) = − 4 πC$_{F}$ α$_{V}$ ( q ) /q 2 in the nonrelativistic limit and extract the coefficient of 1 /r to define the coupling α$_{V}$ ( r ) in the V scheme. The couplings are related by the Fourier transform [60] From (9) we find the expression where C = α$_{V}$ ( Q = 0) = α$_{V}$ ( r → ∞ ) since erf(x → + ∞ ) = 1. We have written explicitly the normalization at Q = 0 in the V scheme since it is not expected to be equal to the normalization in the g$_{1}$ scheme for the reasons discussed in Sec. VII. The couplings in the V and g$_{1}$ schemes are related at leading twist by the CSR: [10] with Q ∗ = 1 . 18 Q , Q ∗∗ = 2 . 73 Q , and we set Q ∗∗∗ = Q $^{∗∗}$. We have verified that this relation numerically holds at least down to Q 2 = 0 . 6 GeV $^{2}$, as shown in the figure in the Appendix (Fig. 7). In order to transform α$_{g}$$_{1}$ ( Q $^{2}$) into α$_{V}$ ( Q $^{2}$) over the full Q 2 range, we extrapolate the CSR to the nonperturbative domain. For guidance, we use the fact that QCD is near conformal at very small Q ; thus, the ratio α$_{V}$ /α$_{g}$$_{1}$ is Q independent. A model for the ratio α$_{V}$ ( Q ) /α$_{g}$$_{1}$ ( Q ) is shown in Fig. 4. We apply this ratio to α AdS Modified,g $_{1}$( Q ), Eq. (11), and then Fourier transform the result using Eq. (17) to obtain α AdS $_{Modified,V}$( r ). We find C ≃ 2 . 2. The right panel of Fig. 5 displays α AdS V ( r ) (dashed line) and α$_{V}$ ( r ) obtained with the same procedure but applied to the JLab data (lower cross-hatched band). Also shown for α$_{V}$ ( r ) = 2 π ∫ ∞ 0 dq α$_{V}$ ( q ) sin( qr ) q . (17) α AdS V ( r ) = C erf( κr ) = 2 √$_{π}$ C ∫ κr 0 e − t $^{2}$dt, (18) α$_{V}$ ( Q $^{2}$) π = α$_{g}$$_{1}$ ( Q ∗ $^{2}$) π − 1 . 09 ( α$_{g}$$_{1}$ ( Q ∗∗ $^{2}$) π ) 2 + 25 . 6 ( α$_{g}$$_{1}$ ( Q ∗∗ $^{2}$) π ) 3 + · · · , (19) 18