UNIFORM POSITION ALIGNMENT ESTIMATE OF SPHERICAL FLOCKING MODEL WITH INTER-PARTICLE BONDING FORCES

The paper proves exponential rendezvous and time-asymptotic flocking on S^2 for the bonded Cucker–Smale system (1.1) with small initial diameters and velocities, via a clean reduction to a linear ODE system dX/dt = A X + F with a negative definite A and rigorously controlled inhomogeneities, plus an exact energy dissipation identity for E = EK + EC; all constants depend only on ψ and σ, not on N. This is explicit in Theorem 1 and the constructions of Sections 2–4, including Proposition 2.7 (energy dissipation), the properties of the rotation R (Proposition 2.4), the linearized system (Proposition 3.1), and the velocity bound (Lemma 4.1). These yield exponential decay of diameters Dx, Dv and the flocking limit as defined in Definition 1.2 . By contrast, the candidate solution contains critical inaccuracies: (i) the stated “modified energy” uses EK + ½ EC whereas the paper’s exact identity holds for EK + EC; the candidate’s factor ½ is unnecessary and misstates the dissipation factor (compare with Proposition 2.7) . (ii) The key second-derivative identity for s_{ij} = ½||xi − xj||^2 has factor-of-two errors: the curvature and bonding contributions should be −s_{ij}(|vi|^2+|vj|^2) and −σ s_{ij} (xi + xj)·m, respectively, not −2 times those terms. (iii) Several steps rely on a Lipschitz bound for R on a cap, but cap-invariance is assumed “since distances shrink,” creating a circular argument; the paper avoids this via the linear ODE framework with A negative definite and a priori uniform bounds on V(t) from energy dissipation . (iv) The z_{ij}-dynamics used to claim exponential velocity alignment is only sketched and lacks a closed, coercive differential inequality without further nontrivial estimates. Consequently, while the final conclusion matches the paper’s theorem, the candidate proof is not correct as written.