Let T be a set of conjugacy classes of Sn such that PT = ∑µ∈T pµ is Schur-positive, and let ΦT be the corresponding genuine character. We record the exact representation-ring matrix attached to this positivity. Multiplication by [ΦT ] is symmetric, positive semidefinite, entrywise nonnegative and integral; its class eigenvalues are zµ on T and 0 off T, so its rank is |T|. More generally, for every irreducible χλ and j ≥ 1, entrywise nonnegativity of powers of the multiplication matrix for ΦT χλ gives ∑µ∈T zµj−1 χλ(µ)j χα(µ)χβ(µ) ≥ 0 for all α, β ⊢ n. Specializing to Sundaram’s Schur-positive set of even non-split classes makes the j = 1 inequality completely unweighted and forces the matrices through the partition-conjugation invariant quotient. We distinguish throughout the classical character-table diagonalization and Sundaram’s Schur-positivity input from the constituent-level inequalities obtained by retaining individual matrix entries. Exact rank and kernel formulas and explicit S3 and S4 examples are included. We further prove a strict-positivity theorem: whenever the selector character ΦT contains every irreducible, every unweighted constituent-level triple sum is strictly positive. Applied to Sundaram’s even non-split selector, this yields strict positivity for all triples of irreducible characters. We conclude with a quantitative lower-bound conjecture supported by exact computation through n = 13.