  The free energy of TAP-solutions for the SK-model of mean field spin glasses
can be expressed as a nonlinear functional of local terms: we exploit this
feature in order to contrive abstract REM-like models which we then solve by a
classical large deviations treatment. This allows to identify the origin of the
physically unsettling quadratic (in the inverse of temperature) correction to
the Parisi free energy for the SK-model, and formalizes the $\textit{true}$
cavity dynamics which acts on TAP-space, i.e. on the space of TAP-solutions.
From a non-spin glass point of view, this work is the first in a series of
refinements which addresses the stability of hierarchical structures in models
of evolving populations.
