OPAC-038: Extremal Negativity in Symmetric-Group Character Tables

This preprint studies OPAC-038, the open problem asking whether, for all sufficiently large (n), the sign representation of the symmetric group (S_n) maximizes the number of negative entries among irreducible character-table rows. The paper develops a collection of exact reductions and structural results aimed at the unweighted nature of the problem, where conjugacy classes are counted once regardless of class size. It proves an exact parity-balance reformulation of OPAC-038, hook-support and Durfee-rank localization results, long-cycle Ferrers matching, primitive-idempotent forcing identities, and a critical Jucys–Weingarten rank framework with complementary-nullity and dual-flatness formulations. Further results include an (O(\sqrt n)) localization theorem for parity-reversing Jucys layers and exact rising-factorial character filters that convert negative character values into nullities of positive-semidefinite Kronecker fusion matrices. The latter gives a magnitude-free rank formulation of OPAC-038 using genuine representations. The paper also develops additional reductions for self-conjugate representations and parity-sensitive character-support formulations, further isolating the remaining difficulty to the non-self-conjugate sector. Extensive exact and computational checks are included to test the structural identities and the selected-rank conjecture over finite ranges. OPAC-038 remains open. The results are intended to narrow the problem to a precise discrete rank/sign obstruction and provide several representation-theoretic settings in which a complete proof may be sought.