A three-layer asymptotic structure for turbulent pipe flow is proposed revealing, in terms of intermediate variables, the existence of a Reynolds-number-invariant logarithmic region for the streamwise mean velocity and variance. The formulation proposes a local velocity scale (which is not the friction velocity) for the intermediate layer and results in two overlap layers. We find that the near-wall overlap layer is governed by a power law for the pipe for all Reynolds numbers, whereas the log law emerges in the second overlap layer (the inertial sublayer) for sufficiently high Reynolds numbers ($Re_{\tau }$). This provides a theoretical basis for explaining the presence of a power law for the mean velocity at low $Re_{\tau }$ and the coexistence of power and log laws at higher $Re_{\tau }$. The classical von Kármán ($\kappa$) and Townsend–Perry ($A_1$) constants are determined from the intermediate-scaled log-law constants; $\kappa$ shows a weak trend at sufficiently high $Re_{\tau }$ but falls within the commonly accepted uncertainty band, whereas $A_1$ exhibits a systematic Reynolds-number dependence until the largest available $Re_{\tau }$. The key insight emerging from the analysis is that the scale separation between two adjacent layers in the pipe is proportional to $\sqrt {Re_{\tau }}$ (rather than $Re_{\tau }$) and therefore the approach to an asymptotically invariant state can be expected to be slow.