Proof of Lemma 3, p. 14 — with K₂ = M³(M + 1), H₂ = M(M + 1) as in Lemma 2, every plan with q₂ ≠ M costs more than the threshold

Let $M \ge U \ge L > 0$ be integers with $L + U < 2\sqrt{2LU}$, and consider the periodic instance $\mathrm{JRP}_{M,U,L}$ with $K_0 = LU$, $K_1 = 0$, $H_1 = 1$, $K_2 = M^3(M+1)$, $H_2 = M(M+1)$. For all positive integers $q_1, q_2$ with $q_2 \neq M$,

So no plan with $q_2 \ne M$ reaches the threshold $z_{M,L,U}$: item 2's heavy EOQ term pins $q_2 = M$, exactly as in the proof of Lemma 2.

Formalization Note The paper says only that $K_2, H_2$ are "sufficiently large … as in the proof … in Lemma 2". The statement uses Lemma 2's values, so the claim is about one explicit instance.