For an irreducible character χλ of Sn, let Nn−(λ) be the number of conjugacy classes on which χλ is negative, with no class-size weighting. Hopkins’ OPAC-038 asks whether, for all sufficiently large n, the sign representation maximizes Nn−. We prove several exact advances toward this problem and isolate the interfaces that remain open. The strongest family-specific results concern balanced hooks: binary Glaisher compression yields an exact two-scale normal form, an injective positivity theorem above the one-third scale, a complete classification of strict negative sources in the first transition band, parity-complete absorption targets, and an infinite family for which every canonical two-part odd split fails. On the exceptional distinct-odd sector, restriction to An gives exact split-fusion multiplicities aij = (g(λ, γi, γj) + χλ(δi)δij)/2, bij = (g(λ, γi, γj) - χλ(δi)δij)/2, and hence g(λ, γi, γi) ≥ |χλ(δi)|. We then derive OPAC-specific unweighted moment inequalities from conjugation modules and Sundaram’s Schur-positive class sums, including Bj ≥ |Oj| for every j ≥ 0. Matrix and plethystic constructions are retained as supporting infrastructure, but are explicitly separated from the paper’s novelty claims: character-table diagonalization, Parseval, and positivity of multiplication matrices are standard consequences of established theory. Finally, we prove no-go statements excluding several tempting scalar, matching, and generic-matrix shortcuts and formulate the remaining task as a two-interface capacity problem. OPAC-038 remains open.