Certificate equivalence theorem: for integers n,r≥1, c≥2, the eventual relation g_{k+r}(n)=c·g_k(n) (all large k) holds iff some orbit point x=g_K(n) satisfies the finite certificate g_r(x)=c·x and rad(c)|g_j(x) for 0≤j<r; moreover the relation then holds for every k≥K and the set of certificate indices is upward closed (so the least certificate index is the sharp onset). Proof: two-line φ-scaling lemma φ(cm)=cφ(m) ⟺ rad(c)|m plus a well-founded double induction; both directions complete in proof_structural_lemmas.md.