Sharp Finite Zero Bounds for One-Parameter Separated Polynomial Systems

Let k >= 2 be even. We determine the largest finite real solution set of a separated system f_i(x_i) = g_i(t), 1 <= i <= m, with all degrees at most k. The maximum is k^2/4 for m = 1, and m(k-1)k^m/4 for m >= 2. The upper bound follows from a weighted count of isolated feasible points of univariate polynomial inequalities: root multiplicity pays for the loss of choices in a fiber at a vanishing constraint. The proof allows constant right-hand sides, multiple roots, and a positive-dimensional complex zero set. For m >= 2, the matching construction is the previously published even-degree family of DannyExperiments (DOI 10.5281/zenodo.21875290); the contribution here is the upper bound throughout the separated class. The unrestricted extremal problem for isolated real zeros of sums of squares is not resolved. The supplement contains exact Python and C++ checks. This is an unrefereed preprint, not a formal certification or an absolute priority claim.