$A_1$ and $B_1$, also called the Townsend-Perry constants, were first measured by Perry and Chong [@PC82; @P86].

More recently, the log-law was generalised to any moment of the streamwise velocity fluctuations, $\tilde w$, assuming Gaussian velocity fluctuations [@MM13], $$\label{eq:l-pfluct}
\langle \tilde w^{2p} \rangle^{1/p} = - A_p \log(\tilde y) + B_p.$$ While the generalised log-law is supported by wall-turbulence experiments, the dependance of $A_p$ and $B_p$ on $p$ turns out to be sub-Gaussian, which is confirmed both experimentally and numerically, [@MM13]. The sub-Gaussian behavior was explained in Ref. [@BC16] using the stochastic closure theory of turbulence [@BB211; @BB314] and the analysis was improved in Ref. [@KBK19], using measurements from the Flow Physics Facility (FPF) at the University of New Hampshire. Both of these studies used the results from homogeneous turbulence [@KBBS17] and made an assumption about the form of the fluctuating shear stress in the inertial layer, based on physical principles.

In Ref. [@GGGC10], a spectral theory for the log-law of the wall of the MVP was proposed in which it is possible to derive the log-law in the inertial layer and the laminar profile in the viscous layer. The novel contribution is the precise form of the transition in the 