We develop a matrix-test framework for \(C^*\) -convex families of completely positive maps \(\textrm{CP}(\mathscr {S},\mathscr {T})\) , where \(\mathscr {S}\) is an operator system and \(\mathscr {T}\) is a unital \(C^*\) -algebra. Matrix tests (k, f, s) induce evaluation functionals \(\Phi \mapsto f(\Phi _k(s))\) and generate a natural weak topology \(\tau \) on \(\mathcal {E}=\operatorname {span}_{\mathbb {C}}(\textrm{CP}(\mathscr {S},\mathscr {T}))\) . Our main result provides a support-function/separation characterization of the \(\tau \) -closure \(\overline{\textrm{cconv}(\mathcal {K})}^{\tau }\) of the \(C^*\) -convex hull of a family \(\mathcal {K}\subseteq \textrm{CP}(\mathscr {S},\mathscr {T})\) in terms of matrix-test inequalities. A key technical tool is a folding lemma that represents every nonzero finite complex linear combination of test functionals, up to a positive scalar factor, by a single matrix test at a suitable block level. As consequences, we obtain a single-test witness for non-membership, support-function criteria for inclusion and equality of the corresponding \(\tau \) -closed hulls, and, under \(0\in \overline{\textrm{cconv}(\mathcal {K})}^{\tau }\) , an exact normalized bipolar-type reconstruction formula. We also show that \(\tau \) is already generated by level-1 tests.
