Let N−n (λ) denote the number of conjugacy classes of the symmetric group Sn on which the irreducible character χλ is negative, with classes counted without size weights. Hopkins’ problem OPAC-038 asks whether, for all sufficiently large n, this statistic is maximized by the sign representation. The global problem remains open. We develop a direct binary-compression route for the balanced hook λk = (k + 1, 1k−1) ⊢ 2k. The hook generating function and the Glaisher split–merge graph reduce the parity-balance deficit to a count of canonical binary edges whose two endpoint values are simultaneously negative versus simultaneously nonnegative. For a canonical edge A = τ ∪ (m, m), B = τ ∪ (2m), with q = k − m and d = k − 2m, we obtain the exact two-scale formula χ(A) = ε(τ )Hd(τ ) − 2(−1)mHq−1(τ ) + Hd−1(τ ), χ(B) = ε(τ )Hd(τ ) − Hd−1(τ ). When the terminal scale m is odd, every strict negative–negative edge with m > d has exactly one higher even part 2b, and the same-scale split 2b 7→ b + b sends its endpoint pair from (−D, −P) to (P + 4C, D) with C ≥ 0. Hence every terminal odd scale m > k/3 is paid injectively, in all degrees and without an asymptotic argument. We then treat the first open band k/4 < m ≤ k/3. Every strict negative–negative edge is proved to have exactly one even part 2b, except for the isolated family τ = (2b, 2b), which has value pair (−2, −2) and is paid explicitly by τ+ = (2b, b, b) with value pair (0, 0). For every remaining one-even source τ = σ ∪ (2b) we derive an exact finite-difference normal form and construct an explicit family of all-odd zero targets ZR = R ∪ (2q − wt(R)), for odd-cardinality submultisets R ⊆ σ of weight below d. Thus every source in the first transition band has at least one favorable target; the unresolved issue is only collision control. We exhibit exact counterexamples showing that odd splits alone and zero absorption alone are each insufficient, and formulate the resulting two-tier split-or-absorb Hall problem. Exact integer computation verifies that hybrid architecture through odd m ≤ 23 on 14,837 one-even negative–negative sources. We also record an exact mixed binary-square identity that identifies the natural aggregate mechanism for the still-open even 2-adic scales.