 Forbidden-intersection density increment

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 $0<\delta \le 1/10$, 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.