 Finite-tuple weak-field property of the generic integral domain

In the generic domain, negation of simultaneous invertibility of finitely many elements implies that one is zero. At a ring stage, localize at their product $t$. All become units, so a tuple satisfying the negation makes the localized stage empty. The empty-cover criterion for the domain-classifying site then says $A[1/t]=0$, hence $t$ is nilpotent. Repeated zero-product covers force one factor to vanish locally. The two-variable property conversely forces the integral-domain axiom in any nontrivial internal ring.