Finiteness of coefficient-polynomial roots

Let $r$ be a nonzero element of the coefficient ring $\mathrm{XCoeff}$ ($r \ne 0$). Then its real root set is finite:

This is the univariate fact (a nonzero real polynomial has finitely many roots, transported across XCoeffEquiv) that makes every 'bad locus' argument work: resultants, leading coefficients, and discriminants each vanish at only finitely many coefficient values, so all but finitely many fibers are good.