Bounded confidence dynamics and graph control: enforcing consensus

The paper’s main consensus theorem for NOLB (r*<1, connected initial data) matches the candidate’s claim, but both arguments have gaps. In the paper, the proof of the key monotonicity claim on pairwise distances in the critical annulus (Proposition 2) appears to conflate cases and implicitly use a property that need not hold (it asserts both inner products are nonnegative when, in general, only one agent may see the other in its critical region). This underpins the corollaries on edge and connectivity preservation and thus the consensus proof, leaving a logical gap. The candidate solution replaces the paper’s route with a consensus-theorems approach (Moreau; Hendrickx–Tsitsiklis) but does not justify that the projected dynamics’ effective weights b_ij(t) satisfy the needed cut-balance/type-symmetry and uniform-boundedness hypotheses; it also uses an unproven bound in the critical annulus. Hence, both are incomplete.