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 (Reτ ). This provides a theoretical basis for explaining the presence of a power law for the mean velocity at low Reτ and the co-existence of power and log laws at higher Reτ . The classical von K´arm´an (κ) and Townsend-Perry (A1) constants are determined from the intermediate-scaled log-law constants; κ shows a weak trend at sufficiently high Reτ but falls within the commonly accepted uncertainty band, whereas A1 exhibits a systematic Reynolds-number dependence until the largest available Reτ . The key insight emerging from the analysis is that the scale separation between two adjacent layers in the pipe is
proportional to √Reτ (rather than Reτ ) and therefore the approach to an asymptotically invariant state can be expected to be slow.