Minimal binary Freiman models have full fourfold sumset ($4A={\mathbb{F}}_2^m,2^m\le K^4|A|$)

If a Freiman 2-isomorphism into ${\mathbb{F}}_2^m$ uses minimal $m$, every nonzero vector lies in the fourfold sumset. Otherwise quotienting by the line it spans would preserve all additive quadruple relations and would still be injective, contradicting minimality. Since zero also belongs to the fourfold sumset, that sumset is the full model group. Plünnecke-Ruzsa inequality then bounds its size by $K^4|A|$. A finite subset of the direct sum of countably many binary fields has a finite-dimensional model to start with.