Strictification

Choose an $R$-dense open $U\subseteq X^2$ where $m$ is defined and both universal translations $\mathrm{\Phi }(x,y)=(x,m(x,y))$ and $\mathrm{\Psi }(x,y)=(m(x,y),y)$ are open immersions; birationality provides such a common domain by intersecting domains of the maps and their inverses. Set $V=\mathrm{\Phi }(U)$, $W=\mathrm{\Psi }(U)$ and $Z=U\cap V\cap W$, all $R$-dense. For each projection $p_i:X^2\to X$, let $T_i$ be its constructible bad-density locus for $Z$, supplied by [F1]. Every generic point of every generic or special fibre of $X$ lies outside $T_i$, since $Z$ contains the generic points of all product-fibre components.

The closure of each $T_i$ omits every fibre generic point. On the generic fibre this follows from constructibility and the first assertion of [F2]. Its generic part has a reduced schematic closure whose ideal is saturated under multiplication by $\pi$. At a special-fibre generic point $\xi$, the local ring is a DVR by [F2]. A nonzero proper ideal of that DVR cannot be $\pi$-saturated: divide an element ${\pi }^{n}u$ repeatedly to obtain a unit. The localized closure ideal is nonzero because the generic bad locus is nondense in its integral component. Hence this generic-part closure misses $\xi$. The special part is constructible within $X_k$ and omits its generic points; [F2] makes its closure nondense there as well. Thus $Q_i=X\setminus \overline{T_i}$ is an $R$-dense open. Set $X^0=Q_1\cap Q_2$; over it $Z$ is dense along both projections.

Define $U^0=U\cap (X^0{\times }_R{X}^0)\cap m^{-1}(X^0)$, with images $V^0=\mathrm{\Phi }(U^0)$ and $W^0=\mathrm{\Psi }(U^0)$ in $(X^0{)}^2$. To check density, base change to a field and fix a point $a\in X^0$. The translation $\mathrm{\Phi }(a,-)$ is an open immersion with dense image in the fibre; intersecting that image with $V\cap (a\times X^0)$ remains dense. Its inverse image imposes $m(a,-)\in X^0$, so $U^0$ is dense along $p_1$. The same argument with $\mathrm{\Psi }(-,a)$ proves density along $p_2$. Since $\mathrm{\Phi }$ preserves $p_1$ and $\mathrm{\Psi }$ preserves $p_2$, it also proves $V^0$ dense along $p_1$ and $W^0$ along $p_2$.

For the remaining densities fix $a\in X^0$ and put $U_a=m^{-1}(a)\subseteq U$ over the chosen field. The open immersions $\mathrm{\Phi }$ and $\mathrm{\Psi }$ identify $U_a$ respectively with $V\cap (X\times a)$ and $W\cap (a\times X)$, dense opens by the construction of $X^0$. Requiring both input coordinates to lie in $X^0$ cuts two dense opens of $U_a$, so $U_a\cap U^0$ is dense in $U_a$. Its $\mathrm{\Psi }$-image is $W^0\cap (a\times X^0)$, dense along the first projection; its $\mathrm{\Phi }$-image is $V^0\cap (X^0\times a)$, dense along the second. All domains and images therefore have both-projection density.

The restricted rational law on $X^0$ is associative by its agreement with the original law on schematic dense domains. Its universal translations are the open immersions above, and both-projection density remains schematic density after arbitrary coordinate base change in the smooth family by [F3]. Thus they give the required test-birational translations and cancellation. The open $X^0$ is smooth, separated and finite type; it meets every special-fibre component and the generic fibre, so it is surjective and flat over $R$, hence faithfully flat. If $X_K$ already carries an everywhere-defined group law, choose $U$ to include $(X_K{)}^2$, where both universal translations are isomorphisms. The generic bad loci are then empty, so $X_K^0=X_K$. This is the BLR model-open strictification needed by completion.