Harmonic-Mode Branch for Topological Memory

Use this decomposition in a graph neural network, mesh encoder, neural state-space model, or long-horizon recurrent predictor when the graph or cell complex has nontrivial first cohomology. Compute an orthonormal basis $H$ for $\ker L$ with an eigensolver, then form $\Pi_H=HH^\top$ and $\Pi_\perp=I-\Pi_H$. Encode every latent edge state as $h=\Pi_Hz$ and $u=\Pi_\perp z$. At each transition, calculate $u\leftarrow u+\delta[-\nu Lu+N_\theta(u,h)]$, and set $h\leftarrow h$ for invariant memory. If changing topological memory is required, replace this with a bounded low-dimensional update $h\leftarrow h+\delta g_\theta(h)$ while keeping $g_\theta$ separate from the dissipative branch. Enforce harmonic invariance by replacing any interaction with $\Pi_\perp N_\theta$, and optionally impose energy-skew structure on $N_\theta$. Compute the harmonic basis and positive spectral gap from the complex; empirically monitor $\|\Pi_H(z_{t+1}-z_t)\|$. The first experiment should use disk, annulus, and two-hole complexes with trajectories initialized at different harmonic circulations, comparing a standard GNN, a fully damped Hodge GNN, and this split model. The prediction is that persistent coordinates equal $\dim\ker L$, distinct harmonic initializations remain separated, harmonic drift stays at numerical precision, and the dissipative residual decays with slope approximately $-\nu\lambda_1^+$.