  We propose a theory for predicting the mean velocity and Reynolds shear and
normal stresses profiles in the wake region of equilibrium adverse
pressure-gradient (PG, APG) turbulent boundary layers (TBLs). Firstly, we
explore the PG-induced dilation-symmetry-breaking of the total stress $\tau^+$
to construct a modified defect power law for $\tau^+$. Crucially, a PG stress
$P_0^+$ is identified, which quantifies the APG-induced total-stress overshoot
and is proportional to the Clauser PG parameter $\beta$. The wall-normal
location with peak stress is predicted. The total stress profiles with
arbitrary $\beta$ are transformed into an invariant profile, which is the
ultimate state of the total stress at infinite $\beta$. This transformation is
equivalent to the outer scaling of the Reynolds shear stress recently-proposed
by Wei & Knopp (JFM, 2023). The Reynolds normal stresses are predicted
accordingly based on the similarity of the Reynolds shear and normal stresses
in the wake region. Secondly, a defect power law is proposed for the stress and
kinetic energy lengths in the wake region. Two critical parameters in the
defect power law are identified to depend on $\beta$ and determine the length
profiles. With the total stress and stress length models, the streamwise
mean-velocity profile is predicted. Especially, an invariant mean velocity
profile is derived, which describes the ultimate state of the mean velocity in
the wake region at infinite $\beta$. This invariant profile is also equivalent
to the outer scaling of Wei & Knopp. The theory also predicts the variation of
the Coles' wake parameter $\Pi$ with $\beta$, in close agreement with the
empirical relation that correlates hundreds of experimental data. The
predictions are validated with five published DNS, LES, and experimental
databases on the equilibrium APG TBLs.
