Cusp Resolution by δ-Coupling for Nonlinear Schrödinger Standing Waves on Compact Quantum Graphs A Universal First-Order Law and Topological Corrections

We identify a universal unfolding law governing a class of degenerate cusp-fold singularities for the nonlinear Schrödinger equation iu_t = -u_xx - f(u) on a compact metric graph G with δ-couplings of strengths α = (α_j) at interior vertices and Neumann conditions at free endpoints. For a broad class of nonlinearities f admitting a degenerate critical point (φ_, ω_) at the cusp of the effective potential, the constant function φ(x) ≡ φ_* is an exact solution at (ω_*, 0), and at this point the linearization L_+ collapses exactly to the pure Kirchhoff Laplacian -∂²_x on G. Under a non-degeneracy hypothesis on f''(φ_*), there is a C¹-smooth codimension-one submanifold Σ ⊂ ℝ × ℝ^|V_int(G)| along which two branches of positive standing waves collide in a saddle-node, with expansion ω = ω_* + (1/|G|) Σ_j α_j + Q_G(α) + O(|α|³) where Q_G is a quadratic form in α. The first-order coefficient 1/|G| is universal: it depends only on the total length and is insensitive to topology, vertex degree, and the distribution of the charges — only Σ_j α_j enters. Topology becomes visible only through Q_G, which in the single-charge case reduces to c_2(G) α²; we derive c_2 explicitly for the lollipop and the figure-eight and verify numerically. Orbital stability follows from the Grillakis–Shatah–Strauss criterion. The universal unfolding is rigid under geometric deformations but is destroyed by structural perturbations eliminating the constant cusp (δ'-coupling, Robin/Dirichlet, magnetic flux). Status This manuscript has been submitted to Nonlinearity (IOP Publishing & London Mathematical Society) on 26 May 2026 (Manuscript ID: NON-110862) and is currently under peer review. This Zenodo deposit serves as a permanent public timestamp of the work and a citable preprint.