Exponential Synchronization of 2D Cellular Neural Networks with Boundary Feedback

The paper’s main theorem (Theorem 3.1) assumes a strictly positive lower bound on the boundary gap signal B(t): lim inf_{t→∞} B(t) > Q[1 + (δ + γ + 2|c − b|)/p] (eq. (3.5)), and concludes full synchronization, i.e., the nearest-neighbor differences Γ, Π, V, W all vanish (eq. (3.6)) . But B(t) is the sum of selected boundary entries of |Γ|^2 and |Π|^2 (specifically i=1 and k=1 via periodicity), so B(t) ≤ Σ(|Γ|^2 + |Π|^2) ≤ Σ(|Γ|^2 + |Π|^2 + |V|^2 + |W|^2). Hence the conclusion lim_{t→∞}(Σ(|Γ|^2 + |Π|^2 + |V|^2 + |W|^2)) = 0 forces B(t) → 0, contradicting the hypothesis that lim inf B(t) is strictly positive. The internal proof mechanics confirm this usage of B(t): inequality (3.17) is combined with (3.5) to get (3.18)–(3.20), showing exponential decay of Γ, Π, V, W; but this same decay would drive B(t) to 0, in conflict with (3.5) holding eventually for all t ≥ T(x0,y0) . Therefore, the theorem as written is logically inconsistent. The candidate solution correctly identifies this contradiction. However, the candidate’s proposed “corrected” proof contains a computational inaccuracy about the control term (they claim an exact identity that the paper only achieves as an inequality with additional positive terms, cf. (3.13)–(3.14)), so the proposed replacement theorem is not established by their sketch .