Mixed Hegselmann-Krause Dynamics II

The uploaded paper (Mixed Hegselmann–Krause Dynamics II, arXiv:2112.13348) states Theorem 5: if 0 ≤ limsup_{t→∞} sup{α_i(t): i∈[n], α_i(t)<1} < 1 almost surely, then τ_δ < ∞ almost surely for all δ>0, where τ_δ is the first time all profile components G̃(t)∩G(t) are δ-trivial. The paper’s proof uses a Lyapunov function Z(t) and a spectral lower bound (Lemma 13) for the sum of squared movements when a component is δ-nontrivial, then splits into group vs. pair interactions. However, in the pair-interaction case the paper writes n^2 ε^2 > 4∑_k (1−γ)δ/2 = ∞ (dimensionally inconsistent and missing squares), which cannot follow from the earlier squared-movement inequality (2) and appears to be a hard error . In the group-interaction case, the proof implicitly relies on times when all agents in the δ-nontrivial component have α_i(t)<1 so that Lemma 13 applies, but this is not justified from the stated assumptions (the limsup condition bounds only those α_i(t) that are <1 and does not ensure “all α_i(t)<1” on infinitely many steps). The argument also conditions on U_t in a way that is unnecessary when Ẽ(t)=E(t) for group interaction, and it asserts “α_i(t)<1 for all i and t during (3)” without being implied by the model’s assumptions . By contrast, the candidate solution reformulates the process on a tail as a bidirectional ρ-agreement system (uniform ρ>0 extracted from the limsup assumption) and invokes Chazelle’s finite total s-energy result to show that only finitely many times can have an edge longer than ε; then, by a path/triangle inequality, all components become δ-trivial after some finite time. This proof is concise, avoids the paper’s gaps, and is correct under the given assumptions (it uses only limsup<1 and undirected profile graphs) while handling α_i(t)=1 gracefully by adding self-loops to the neighbor sets.