Regularisation of isolated codimension-2 discontinuity sets

The paper’s Theorem 4.1 establishes exactly the two claims the candidate addresses: for the layer problem in the scaling chart κ2, there is a unique equilibrium iff (A(z)^{-1}f(0,0,z))^T A(z)^{-1}f(0,0,z) < 1 and none if it is > 1, and when the equilibrium exists the reduced slow flow is ż = -B(z)A(z)^{-1}f(0,0,z) + g(0,0,z) (equations (4.14)–(4.17)) . The paper’s proof sketches existence by invoking e_Ψ^T e_Ψ < 1 and uniqueness via Lemma 2.6(c) (one-to-one) , relying implicitly on the radial structure e_Ψ(l cosθ, l sinθ) = L_Ψ(l)(cosθ, sinθ) from Remark 2.5 and the regularisation assumptions (Definition 2.3) . The candidate solution supplies the missing explicit step: it proves L_Ψ is C^1, strictly increasing, and maps [0,∞) bijectively onto [0,1), hence e_Ψ: R^2 → D_1 is onto and injective. This uses precisely the paper’s assumptions Ψ(0)=1, Ψ>0, Ψ'≤0 (Definition 2.3) and rotation equivariance (Lemma 2.6(a)) . Substitution into the slow equation yields ż = -B A^{-1} f + g, matching (4.17) and the Filippov correspondence noted in the paper . On the boundary case (A^{-1}f)^T(A^{-1}f)=1, the paper treats this as a bifurcation where the critical set is created “at infinity,” so no finite equilibrium exists in κ2 at equality, consistent with the candidate’s observation that the image of e_Ψ(·;1) is the open unit disc .