Parity theorem: for m≥3, g(m)≡m (mod 2), so orbit parity is constant once the orbit is ≥3; consequently any eventual relation with even multiplier c (in particular the literal Erdős–Graham c=2) forces n even (n≥4) with an all-even orbit, and odd n≥3 admit only odd multipliers. All odd witnesses in the catalogue (6075, 9009, 11739, 13857, 31851, 74829, 965505) have odd c ∈ {729, 6561}, as forced.