Conserved charges in linearly reinforced graph urns

In the graph-based Pólya urn with linear reinforcement (WARM with $\alpha=1$), every vector $\sigma$ in the kernel of the unsigned vertex–edge incidence matrix yields an exact martingale $\sum_e\sigma_e H(N_e-1)$, where $H$ is the harmonic number. These charges are the flat directions of the Lyapunov function, and they determine which equilibrium the process selects. On the even cycle $C_{L}$ the limit is $(1+u,1-u,\dots)$ with $L\,\mathrm{artanh}(u)$ equal to the terminal value of the charge, and $\sqrt{L}\,\mathrm{artanh}(u)$ converges in law to a centred Gaussian as $L\to\infty$. For large even cycles this describes the law of the random limit, in the continuous-time form of the question raised by Chen and Lucas and by Lima for the discrete-time urn.