  The spin glass behavior near zero temperature is a complicated matter. To get
an easier access to the spin glass order parameter $Q(x)$ and, at the same
time, keep track of $Q_{ab}$, its matrix aspect, and hence of the Hessian
controlling stability, we investigate an expansion of the replicated free
energy functional around its ``spherical'' approximation. This expansion is
obtained by introducing a constraint-field and a (double) Legendre Transform
expressed in terms of spin correlators and constraint-field correlators. The
spherical approximation has the spin fluctuations treated with a global
constraint and the expansion of the Legendre Transformed functional brings them
closer and closer to the Ising local constraint. In this paper we examine the
first contribution of the systematic corrections to the spherical starting
point.

Disordered Systems and Neural Networks