  We study the high-temperature regime of a mean-field spin glass model whose
couplings matrix is orthogonally invariant in law. The magnetization of this
model is conjectured to satisfy a system of TAP equations, originally derived
by Parisi and Potters using a diagrammatic expansion of the Gibbs free energy.
We prove that this TAP description is correct in an $L^2$ sense, in a regime of
sufficiently high temperature. Our approach develops a novel geometric argument
for proving the convergence of an Approximate Message Passing (AMP) algorithm
to the magnetization vector, which is applicable in models without i.i.d.
couplings. This convergence is shown via a conditional second moment analysis
of the free energy restricted to a thin band around the output of the AMP
algorithm, in a system of many "orthogonal" replicas.
