Quantitative forbidden-intersection bound by widening ($\rho \le \max \{e^{-g\ell +D^{2}n/4},e^{-gn+(2g+h)\ell }\}$)

Let two set families on $n$ coordinates forbid cross-intersection size $\ell$, and let $\rho$ be their density product. For $0<\delta \le 1/10$, put $g=\log (1+\delta )$, $h=-\log (1-\delta -2{\delta }^2)$, and $D=g+h$. Repeated forbidden-intersection density increments terminate when one endpoint of the forbidden interval reaches zero or the remaining dimension. In the first case, if $v$ steps widened the interval, the cross-intersection bound from cube separation gives $\log \rho \le -g\ell +Dv-v^2/n\le -g\ell +D^{2}n/4$. In the second case, at least $n-\ell$ steps occurred and at most $\ell$ widened, giving $\log \rho \le -gn+(2g+h)\ell$. For a single set family take the square of its density. Choosing $\delta =1/50$ proves $|\mathcal{A}|\le 1.999^n$ for forbidden intersections $n/4$ and $\lfloor n/8\rfloor$, with small dimensions handled directly.