On the characterization of models of H* : The operational aspect

We give a characterization, with respect to a large class of models of untyped $\lambda$-calculus, of those models that are fully abstract for head-normalization, i.e., whose equational theory is ${\mathcal{H}}^{\ast }$. An extensional K-model $D$ is fully abstract if and only if it is hyperimmune, i.e., non-well founded chains of elements of $D$ cannot be captured by any recursive function. This article share its first title with its companion paper and a short version. It is a standalone paper that present a purely syntactical proof of the result as opposed to its companion paper that present an independent and purely semantical proof of the exact same result.