Consensus Dynamics of the D-Operator: Convergence, Local Stability, and the Structure of Non-Consensus Equilibria

The D-Operator is a vector generalization of the Kuramoto model, previously verified only through scenario-specific numerical checks (e.g. Checks A–D of the PhaseSync-on-a-D-Operator-Core paper, which already exercised the κ=0/κ>0 contrast this paper analyzes). This paper replaces those scenario-specific observations with closed-form or general results, organized by which parameter is active. In the linear regime (λ=0), we derive the exact mode decomposition of the error dynamics and show that a previously reported result -- "stronger coupling accelerates convergence" -- holds only transiently: the asymptotic rate toward a shared external target is governed by the anchoring strength γ alone, independent of coupling. In the nonlinear regime (λ>0), we show the dynamics is a gradient flow for symmetric coupling, extend this to the full three-term regime, and derive a closed-form resolution of a previously only qualitatively described "tug-of-war" phenomenon. We then prove local exponential stability of the consensus manifold as a general theorem -- for any connected symmetric or strongly connected directed graph and any positive λ, κ -- via Hessian and Jacobian eigenvalue analysis mirroring the Master Stability Function technique used in the high-dimensional Kuramoto literature. Finally, we show that the strongest possible claim, unqualified global convergence, is literally false: non-consensus equilibria exist (an explicit closed form for n=2 and a generalization to n-fold splay states for complete graphs), but every one found -- analytically or via numerical search on directed graphs -- is an unstable saddle point, refining "always converges" into the precise and defensible claim that consensus is the unique generic attractor.