  A single McCulloch-Pitts neuron, that is, the simple perceptron is studied,
with focus on the region beyond storage capacity. It is shown that Parisi's
hierarchical ansatz for the overlap matrix of the synaptic couplings with so
called continuous replica symmetry breaking is a solution, and as we propose it
is the exact one, to the equilibrium problem. We describe some of the most
salient features of the theory and give results about the low temperature
region. In particular, the basics of the Parisi technique and the way to
calculate thermodynamical expectation values is explained. We have numerically
extremized the replica free energy functional for some parameter settings, and
thus obtained the order parameter function, i.e., the probability distribution
of overlaps. That enabled us to evaluate the probability density of the local
stability parameter. We also performed a simulation and found a local stability
density closer to the theoretical curve than previous numerical results were.
