PROGRESS ON OPAC-038: BINARY COMPRESSION, SPLIT FUSION, AND UNWEIGHTED CHARACTER INEQUALITIES

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 global results apply to every irreducible row: conjugation modules and Sundaram’s Schur-positive class sums yield unweighted moment domination on parity-balanced class reservoirs. A complementary exact case study treats balanced hooks under Glaisher compression: we derive a closed formula across a canonical Glaisher edge, prove an injective positivity theorem above the threshold m > k/3, classify all strict negative sources in the next range k/4 < m ≤ k/3, construct parity-complete all-odd targets, and exhibit an infinite family for which every canonical two-part odd refinement 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. We end with a precise account of what remains open; the proposed coupling is presented only as a research programme, not as a theorem or a reduction of OPAC-038. OPAC-038 remains open.