A non-Gaussian limit for nodal surplus

For each $m \ge 3$, an explicit connected loop-free quantum graph with cycle rank $\beta_m = 2m - 1$, no vertex of degree two and rationally independent edge lengths, under the standard Kirchhoff Laplacian, whose nodal surplus $S_m$ satisfies $\operatorname{Var}(S_m)/\beta_m \to a_*/2 \in (1/8, 1/4)$ while $(S_m - \beta_m/2)/\sqrt{\operatorname{Var}(S_m)}$ converges to a variance mixture $Z\sqrt{V/a_*}$ of Gaussians with a nonconstant $V$, which is not Gaussian. This disproves universal Gaussian behaviour of the nodal surplus as $\beta \to \infty$; the universal linear variance bounds are left open.