Quantitative Relaxation Dynamics from Generic Initial Configurations in the Inertial Kuramoto Model

The paper’s Theorem 1.1 establishes asymptotic phase-locking for the inertial Kuramoto model under the smallness framework (1.3)–(1.4), and further guarantees a majority cluster with quantitative ordering; the condition (1.4) features exactly the gain with the numerical factor √3.068 that appears in the candidate’s argument. The candidate solution follows a different route: it derives a closed pairwise equation, sets up a two-variable (diameter/velocity) contraction over time-windows, and then uses an energy argument (LaSalle/gradient-flow) to conclude convergence. This aligns with the paper’s outcomes (asymptotic locking and majority cluster with bounds) and even reproduces the precise smallness gain in (1.4) of Theorem 1.1. However, the model’s Step 3 asserts a global forward-invariant cone (all pairwise gaps < π/2) from the windowed contraction; that stronger global bound is not proved in the paper’s route (which instead proves emergence and stability of a majority cluster), and in the model sketch it is not fully justified beyond the contraction inequality. Despite this, the end conclusions of the model agree with the paper’s claims, and the two approaches are substantially different: the paper pieces together an initial-layer control of R and Δ, stability of a majority cluster, and an inertial gradient-flow/Łojasiewicz argument, whereas the model compresses this into a DV-contraction plus convexity/energy. Overall: both correct, by different proofs. Citations: the paper’s Theorem 1.1 and its conditions (1.3)–(1.4) with the √3.068 constant are explicit, as are the mean-rotation identities (Lemma 2.1), the order-parameter formulation (Lemma 3.1), and the inertial gradient-flow/Łojasiewicz framework (Proposition 3.1) used in the paper’s proof path, which culminates via Theorem 3.2 under framework (F) and its construction in Appendix C .