CLEARMIND: Closed-Form Laplacian Embeddings of Arakelov-Green Resistance for Morphological Intrinsic Neuronal Distances

Tropical geometry turns a metric graph into a flat torus, its tropical Jacobian, and measures distances between the images of its points by a closest vector problem on the period lattice. We prove that for points of the graph this problem is solved by geodesics: the squared tropical polarization distance equals the path metric minus the Arakelov–Green distance, an effective-resistance kernel given in closed form by the Laplacian pseudoinverse. Equivalently, the Arakelov–Green distance and the path metric are the minimal energies of real and of integral unit flows, and the squared tropical distance is exactly their integrality gap. The full tropical distance matrix is thus computable in cubic time. On this identity we build CLEARMIND, a training-free descriptor of 3D neuronal morphology. A reconstruction is reduced to its branching skeleton, neurite tips near the soma are joined to it, short bridges are contracted, and the Arakelov–Green matrix of the result is summarized by its spectrum. Every step has an exact algebraic description; for instance, the period matrix records the shared soma-to-tip path lengths, and the spectrum has exactly one positive eigenvalue, so its absolute values determine it. The signature needs no lattice search, is invariant to vertex order, rigid motions and subdivision, and is Lipschitz in the edge lengths. On ACT-4, JML-4 and BIL-6, appending the 64-dimensional spectrum improves a point-cloud GNN and MorphVAE on every dataset, by up to 20.3 points; a Tree-LSTM with the spectrum outperforms every reimplemented baseline; and an MLP on the spectrum alone is a competitive morphology classifier that surpasses every reimplemented deep baseline on ACT-4. On BREC, a GIN with Arakelov–Green eigenvector encodings distinguishes 70.0% of the pairs beyond the 1-WL limit, including 33% of the CFI pairs, more than PPGN (23%) and I${}^2$-GNN (21%), and the spectrum alone separates 217 of the 400 pairs without training.