= Forbidden-intersection density increment
{title2=$[a,b]\mapsto[a-1,b]$}
For two <set families> forbidding all cross-intersections in $[a,b]$, split into absent and present coordinate sections. The section pairs $(\mathcal F_1,\mathcal G_1)$, $(\mathcal F_0,\mathcal G_0\cup\mathcal G_1)$, and $(\mathcal F_1,\mathcal G_0\cap\mathcal G_1)$ forbid, respectively, $[a-1,b-1]$, $[a,b]$, and $[a-1,b]$ in one fewer dimension. For $0<\delta\leq1/10$, either one of the first two pairs, allowing exchange of the <set family> names, increases the density product by at least $1+\delta$, or the third widens the interval and retains at least $1-\delta-2\delta^2$ of that product. This converts a single forbidden intersection into a long forbidden interval while recording every density gain and loss.
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