= Minimal binary Freiman models have full fourfold sumset
{title2=$4A=\mathbb F_2^m,\quad2^m\leq K^4|A|$}
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.
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