For an irreducible character χλ of the symmetric group Sn, let Nn−(λ) = #{µ ⊢ n : χλ(µ) < 0}, with conjugacy classes counted uniformly. OPAC-038 asks whether, for all sufficiently large n, this statistic is maximized by the sign representation. The problem remains open. We give a standalone synthesis of exact structural advances toward the problem. The sign row has (p(n) − q(n))/2 negative entries, where q(n) is the number of self-conjugate partitions, and extremality is equivalent to a parity-balance inequality. A known vanishing theorem for self-conjugate characters implies that self-conjugate rows satisfy the desired bound for all sufficiently large n. An explicit rising-factorial character filter converts the sign problem exactly into a rank/support lower bound for a genuine positive-semidefinite fusion matrix. We identify a canonical three-sector decomposition of the representation ring, of dimensions r, r, q, corresponding to odd classes, even non-distinct-odd classes, and distinct-odd classes. For balanced hooks we derive an exact formula across canonical Glaisher edges, prove injective positivity above the one-third threshold, classify all strict negative sources in the next range, construct parity-complete all-odd absorption targets, and exhibit an infinite family for which every canonical two-part odd refinement fails. On the distinct-odd sector, restriction to An yields exact split-fusion multiplicities and the bound g(λ, γ, γ) ≥ |χλ(δ)|. Conjugation modules and Schur-positive class sums give genuinely unweighted moment domination and constituent-level nonnegative localizers. Finally, the positive filter admits an adaptive oriented-transversal certificate. After passing to conjugation-adapted columns, a Laplace expansion turns this certificate into simultaneous nonvanishing of complementary odd and even character-table minors, equivalently a representable matroid-capacity problem. A fixed orientation is impossible already in S4. This complementary-minor problem is the sharpest reduction obtained here; it is sufficient for OPAC-038 but is not proved. We state all open assertions explicitly and record several no-go results that rule out scalar moments, generic matrix positivity, one-reservoir matching, and antisymmetric-only Kronecker certificates as complete arguments.