Extend the Penrose lower bound to broader intrinsic-frequency distributions

Prove that for any intrinsic-frequency distribution g on R that is the symmetric sum of multiple Dirac measures—and, potentially, for any symmetric distribution g—the inequality κ₂(g) ≥ κ_P(g) holds. Here κ₂(g) denotes the largest coupling strength κ for which the g-uniform equilibrium m⊗g is locally stable in time (as defined in Definition 3), and κ_P(g) is the Penrose threshold defined by κ_P(g) := inf{κ > 0 : ∃ θ ∈ R with P(iθ) = 2/κ}, where P(z) = ∫_R 1/[(γ + iω − z)(σ²/2 + z − iω)] g(dω).

Background

The paper introduces several thresholds governing synchronization and stability in a Kuramoto mean field game with intrinsic frequency distribution g: κ₁(g), κ₂(g), κ_c(g), and the Penrose constant κ_P(g). The authors prove general upper and lower bounds relating κ₁(g), κ₂(g), and κ_c(g), and develop a Laplace-transform-based analysis to relate stability to zeros of a function P(z).

For the specific case where g is the symmetric sum of two Dirac measures, Theorem \ref{th:two_Diracs} establishes the stability lower bound κ₂(g) ≥ κ_P(g). The authors further discuss the Penrose condition and show how it yields invertibility of the linearized operator that underpins stability at the g-uniform equilibrium.

Motivated by this two-Dirac result, the authors explicitly conjecture that the same lower bound should hold more generally, first for symmetric sums of multiple Dirac measures and possibly for any symmetric g. Remark \ref{rem:many} highlights technical obstacles in extending their Laplace-domain approach beyond two Dirac masses, indicating why this remains an unresolved problem.