Discrete-Time Periodic Monotonicity Preserving Systems

The paper’s Theorem 4 (i)⇔(ii)⇔(iii) is explicitly stated and proved in the appendix, using the convexity-preservation equivalence (Lemma 10) and the convex-contour characterization (Theorem 1), plus the oriented-area identity det[[Δg(t),Δg(t−1);Δg(t+1),Δg(t)]] = det[[Δg(t−1),g(t−1),1];[Δg(t),g(t),1];[Δg(t+1),g(t+1),1]] to obtain the log-concavity condition and PM of Δg. This matches (ii)⇔(iii) and supplies a correct proof of (ii)⇒(i) (; ; ; ). By contrast, the model’s key step for (i)⇔(ii) invokes the Karlin–Schoenberg variation‑diminishing characterization to conclude that the circulant Toeplitz matrix K with kernel Δg must be cyclic sign‑regular of order 3. This misapplies the theorem: PMP(T) requires Sg to preserve unimodality of inputs (equivalently S_c[Δu]≤2), not to be CVB2 on all inputs with S_c[u]≤2. The paper itself emphasizes PMP(T) is strictly weaker than CVB2(T) (Lemma 9), and provides a different, correct route from (ii)⇒(i) without assuming sign‑regularity of K (; ).