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Tropical Algebraic Geometry for Neuronal Representations: An Arakelov-Green Measure Based Descriptor for Graph Learning
Yuyang Zhang, Weihan Xu, Xuehai Zhou, Shucheng Cao, Qihuang Zhang
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Abstract
Abstract:The quantitative analysis of 3D neuronal morphologies requires capturing both graph topology and spatial geometry. Current message-passing Graph Neural Networks (GNNs) are bounded by the 1-Weisfeiler-Lehman (1-WL) test, limiting their ability to capture cycles induced by spatial proximities. To address this, we propose a training-free geometric prior based on tropical algebraic geometry. We apply the recently established tropical Abel-Jacobi transform and polarization distances to machine learning on tree-structured data. We introduce a structural transformation pipeline, comprising cycle space augmentation and quotient space construction, to convert spatial trees into cyclic metric graphs suitable for embedding into the Tropical Jacobian. Computing exact tropical polarization distances requires solving the NP-Hard Closest Vector Problem (CVP) on integer lattices. Instead of relying on explicit approximations with quantization errors (e.g., Babai's rounding), we adopt a continuous relaxation on the universal cover of the Albanese torus. We show that the discrete Arakelov-Green measure, computed in closed form via the graph Laplacian's generalized inverse, decomposes exactly into the intrinsic path metric minus the unquantized polarization distance on this cover, avoiding integer lattice searches. This metric yields two descriptors: eigenvectors provide node-level structural coordinates, and the permutation-invariant eigenvalue spectrum provides a graph-level signature. On the BREC benchmark, the eigenvector formulation demonstrates expressivity beyond the 1-WL limit. On 3D morphology datasets (ACT-4, JML-4, BIL-6), the spectrum seamlessly integrates into standard architectures (VAEs, GNNs, Tree-LSTMs) without additional trainable parameters, outperforming explicit lattice approximations and improving classification accuracy over existing spatial models.
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Tropical Algebraic Geometry for Neuronal Representations: An Arakelov-Green Measure Based Descriptor for Graph Learning Yuyang Zhang1, Weihan Xu1, Xuehai Zhou1, Shucheng Cao1, Qihuang Zhang1 Corresponding author Abstract The quantitative analysis of 3D neuronal morphologies requires representations that capture both graph topology and 3D spatial geometry. Current message-passing Graph Neural Networks (GNNs) are bounded by the 1-Weisfeiler-Lehman (1-WL) test, limiting their ability to capture cycles induced by spatial proximities. To address this, we propose a training-free geometric prior based on tropical algebraic geometry. Recently, the computational foundations for the tropical Abel-Jacobi transform and tropical polarization distances were established for metric graphs. In this work, we apply these mathematical tools to practical machine learning tasks for tree-structured data. We introduce a structural transformation pipeline, comprising cycle space augmentation and quotient space construction, to convert spatial trees into cyclic metric graphs suitable for embeddings into the Tropical Jacobian. Computing exact tropical polarization distances requires solving the Closest Vector Problem (CVP) on integer lattices, which is NP-Hard. Instead of relying on explicit lattice approximations with quantization errors (e.g., Babai’s rounding), we adopt a continuous relaxation on the universal cover of the Albanese torus. We show that the discrete Arakelov-Green measure, computed in closed form via the generalized inverse of the graph Laplacian, decomposes exactly into the intrinsic path metric minus the unquantized polarization distance on that cover, and is therefore obtained without any integer lattice search. This metric yields two descriptors: its eigenvectors provide node-level structural coordinates, and its permutation-invariant eigenvalue spectrum provides a graph-level signature. On the BREC benchmark, the eigenvector formulation demonstrates expressivity beyond the 1-WL limit. On 3D morphology datasets (ACT-4, JML-4, BIL-6), the spectrum integrates into standard architectures (VAEs, GNNs, Tree-LSTMs) without additional trainable parameters. It outperforms explicit lattice approximations and improves classification accuracy compared to existing spatial models. Introduction The quantitative analysis of 3D neuronal morphologies requires learning representations from spatial trees, where nodes possess 3D Euclidean coordinates. A primary challenge is constructing representations that capture both the graph topology and the spatial geometry, while maintaining invariance to node permutations. Standard message-passing Graph Neural Networks (GNNs) are bounded by the 1-Weisfeiler-Lehman (1-WL) graph isomorphism test (Xu et al. 2019). In the context of spatial trees, this limits their ability to capture cycles induced by spatially close branches. Conversely, Topological Data Analysis (TDA) captures global cycles but discards pointwise geometric information when vectorizing discrete persistence diagrams into fixed-length descriptors. To evaluate spatial trees using tropical algebraic geometry, we introduce a structural transformation pipeline. Recently, Cao and Monod (Cao and Monod 2025) formalized the computational foundations for the tropical Abel-Jacobi transform and tropical polarization distances on metric graphs. We are the first to apply this mathematical framework to practical machine learning tasks, specifically neuronal morphology classification. We perform graph augmentation by adding edges between the root node and spatially close leaf nodes (under a distance threshold ϵε). This transforms the input tree into a graph with a non-trivial cycle space (first Betti number β1>0 _1>0), enabling the projection of the graph into a flat Riemannian torus, the Tropical Jacobian Jac(Γ)=ℝg/ΛJac( )=R^g/ , via the tropical Abel-Jacobi map. To filter terminal acyclic noise, we apply a quotient space construction prior to distance evaluation. However, utilizing this mathematical framework in large-scale machine learning presents a computational barrier. As proven (Cao and Monod 2025), computing the exact tropical polarization distance on the Tropical Jacobian is mathematically equivalent to the Closest Vector Problem (CVP) on the integer lattice Λ , which is NP-Hard (van Emde Boas 1981b). Existing explicit computations rely on lattice basis reduction algorithms, such as Babai’s rounding(Babai 1986). These discrete approximations introduce quantization errors that degrade the metric space and impose high computational costs. We address this problem by adopting a continuous relaxation on the universal cover ℝgR^g. Utilizing the discrete Hodge orthogonal decomposition, we show that the discrete Arakelov-Green measure, computed in closed form via the Moore-Penrose pseudoinverse (L+L^+) of the discrete Laplace-Beltrami operator, decomposes exactly into the intrinsic path metric minus the unquantized polarization distance on ℝgR^g. By relying on this decomposition, our formulation obtains a robust continuous metric without any integer lattice search, eliminating the quantization errors of discrete approximations. Based on this mechanism, we extract two deterministic, training-free structural descriptors from the Arakelov-Green measure. We summarize our contributions as follows: • Application of the Tropical Jacobian to Neuronal Morphologies: We bridge the mathematical framework of the tropical Abel-Jacobi transform to practical machine learning. We design a graph augmentation and quotient space construction pipeline that converts acyclic spatial trees into cyclic metric graphs, encoding 3D spatial proximities into homology groups while filtering data acquisition artifacts. • Continuous Relaxation vs. Explicit Lattice Approximation: We prove that the discrete Arakelov-Green measure decomposes exactly into the intrinsic path metric minus the unquantized polarization distance on the universal cover ℝgR^g. This continuous relaxation ((|V|3)O(|V|^3)) is obtained in closed form without any integer lattice search, eliminating the quantization errors associated with explicit lattice rounding algorithms. • Theoretical Expressivity and Empirical Performance: We demonstrate that this metric yields two distinct descriptors. On the BREC benchmark (Wang and Zhang 2024), its node-level eigenvector formulation separates graph pairs beyond the 1-WL limit. On 3D morphology datasets (ACT-4, JML-4, BIL-6), its graph-level permutation-invariant eigenvalue spectrum integrates into standard architectures (MLPs, GNNs, Tree-LSTMs) without additional trainable parameters and improves classification accuracy. Related Work Morphological Representations in Connectomics Neuronal morphologies are conventionally modeled as spatial trees. Deep learning approaches include sequence-based, point-cloud-based, and graph-based models. Generative models like MorphVAE (Laturnus and Berens 2021) utilize 3D random walks, but stochastic sampling limits global topology extraction. Point-cloud networks such as MorphoGNN (Zhu et al. 2023) apply spatial convolutions over downsampled coordinates, often discarding exact path distances. Tree-based models, including Tree-LSTMs (Tai et al. 2015) and TreeMoCo (Chen et al. 2022), process morphologies as directed acyclic graphs (β1=0 _1=0), failing to capture cyclic structures formed by spatially close branches. Our framework addresses this by adding edges based on spatial proximity, encoding 3D geometry into the homology groups of the graph. Graph Neural Network Expressivity and Spectral Encodings The discriminative power of standard Message Passing Neural Networks (MPNNs) is bounded by the 1-Weisfeiler-Lehman (1-WL) test (Xu et al. 2019). To improve expressiveness, methods incorporate Positional Encodings (PE) or Structural Encodings (SE) derived from graph Laplacian eigenvectors (Dwivedi et al. 2023). However, explicit spectral features suffer from sign and basis ambiguities, requiring specialized canonization (Ma et al. 2023). Alternatively, encoding the effective resistance metric improves expressiveness beyond 1-WL (Zhang et al. 2023; Klein and Randić 1993). Our approach aligns with this direction using tropical algebraic geometry. We compute the discrete Arakelov-Green measure via the generalized inverse of the discrete Laplace-Beltrami operator (L+L^+), corresponding to the effective resistance on a quotient space. Extracting the eigenvalue spectrum of this matrix yields a permutation-invariant structural signature, bypassing Laplacian eigenvector ambiguities. As validated on the BREC benchmark (Wang and Zhang 2024), this signature demonstrates expressivity beyond standard 1-WL message passing. Topological Data Analysis (TDA) Topological Data Analysis (TDA) extracts shape characteristics invariant to continuous deformations. For neurons, the Topological Morphology Descriptor (TMD) (Kanari et al. 2019) tracks connected components across a radial filtration. Integrating these discrete persistence diagrams into neural networks requires transformation into continuous representations via Gaussian kernel density estimation (e.g., Persistence Images (Adams et al. 2017)). This vectorization process alters the original path distances of the graph and summarizes the structure into a 2D image, preventing exact node-level spatial alignment. Our formulation avoids kernel approximations. The generalized inverse L+L^+ induces a valid Euclidean Distance Matrix (EDM). Extracting the eigenvalue spectrum of this matrix directly encodes the graph geometry into a fixed-length topological signature. Tropical Geometry and Metric Graphs Tropical algebraic geometry analyzes metric graphs by translating Riemann surfaces into tropical curves. Cao and Monod (Cao and Monod 2025) formalize the computation of the tropical Abel-Jacobi transform and polarization distances. They prove that evaluating these distances on the tropical Jacobian requires solving the Closest Vector Problem (CVP) on a lattice, which is NP-Hard. Practical computations thus rely on approximation algorithms like Babai’s nearest plane method. Baker and Faber (Baker and Faber 2010) relate Jacobian metrics to effective resistance, showing that structural bridges collapse to a point under the tropical Abel-Jacobi map. Building on this, we construct a quotient space to contract microscopic bridges. Instead of approximating the NP-hard CVP on the integer lattice, we evaluate the discrete Arakelov-Green measure. Drawing on the formal distinction between the quotient Albanese torus and its universal cover ℝgR^g (Caporaso and Viviani 2010), we show that this measure decomposes in closed form into the intrinsic path metric minus the unquantized polarization distance on the universal cover. This continuous relaxation requires no integer lattice search and incurs no rounding error. Methodology Our objective is to extract a permutation-invariant and SE(3)SE(3)-invariant structural signature from 3D spatial trees. The framework processes the input structure through metric graph transformation, cycle augmentation, and evaluation of the unquantized polarization distance on the universal cover of the Tropical Jacobian. Figure 1 illustrates the structural transformation stages, and Algorithm 1 summarises the complete procedure. Figure 1: Overview of the structural transformation pipeline. (a) The input neuronal morphology is modeled as a spatial tree LCCT_LCC. (b) Kron reduction eliminates valence-2 vertices. The intrinsic path metric is preserved by accumulating the edge lengths (ℓnew=ℓ1+ℓ2 _new= _1+ _2), simplifying the graph topology without discarding metric weights. (c) Cycle space augmentation connects spatially proximate leaves to the root within a threshold ϵε. This transforms the tree into a cyclic graph (β1>0 _1>0), encoding 3D spatial geometry into fundamental homology groups. (d) Quotient space construction contracts terminal cut-edges below a length threshold τ by driving the Laplacian affinities to infinity (Wu,v→∞W_u,v→∞), filtering data acquisition artifacts to construct the final metric graph. Metric Graph Formulation and Topological Retraction Let a 3D spatial tree be denoted by =(V,E)T=(V,E), where V and E are the sets of vertices and edges, respectively. We formulate this structure as an abstract metric graph Γ=(,ℓ) =(T, ). The length function ℓ:E→ℝ>0 :E _>0 is defined as the L2L_2 distance of the Euclidean embedding ι:V→ℝ3 :V ^3. We operate on the largest connected component, LCCT_LCC. Spatial graphs derived from biological data often contain over-sampled linear segments. To make the representation invariant to sampling density, we apply Kron reduction (Dörfler and Bullo 2013) to eliminate degree-2 vertices. Theorem 0.1 (Metric Preservation via Kron Reduction). Let Vreg=v∈V∖vroot∣deg(v)=2V_reg=\v∈ V \v_root\ (v)=2\ be the set of valence-2 regular vertices. Evaluating the Schur complement of the combinatorial graph Laplacian with respect to VregV_reg extracts the minimal skeleton Γmin _ . This operation preserves the discrete Arakelov-Green measure between the retained vertices Vcore=V∖VregV_core=V V_reg. Proof Sketch. Since this reduction operates on the initial tree LCCT_LCC prior to any edge augmentation, the graph contains no cycles. The Schur complement sequentializes the inverse edge weights. This process does not discard metric weights; instead, it accumulates the intrinsic lengths of the eliminated segments into a single consolidated edge (ℓ(u,w)=ℓ(u,v)+ℓ(v,w) (u,w)= (u,v)+ (v,w)). Consequently, the updated off-diagonal entries equal the reciprocal of the sum of the original metric lengths, preserving the exact path distances between all remaining core vertices. (See Appendix A.1 for the full proof). □ This reduction decreases the vertex set size from |V||V| to |Vcore||V_core|, bounding the subsequent computational complexity to (|Vcore|3)O(|V_core|^3). Cycle Space Augmentation According to the Euler-Poincaré formula, the minimal skeleton Γmin _ of a tree has a first Betti number β1=0 _1=0. Consequently, its Jacobian variety is zero-dimensional, meaning the graph topology does not encode extrinsic 3D spatial folding. We introduce cycle space augmentation to encode spatial geometry into the graph topology. Given a distance threshold ϵ>0ε>0, let Vcore, leafV_core, leaf denote the leaf vertices of Γmin _ . We define a set of new edges Eϵ=(v,vroot)∣v∈Vcore, leaf,‖ι(v)−ι(vroot)‖2<ϵE_ε=\(v,v_root) v∈ V_core, leaf,\| (v)- (v_root)\|_2<ε\ and construct the augmented graph Γϵ=Γmin∪Eϵ _ε= _ ∪ E_ε. By adding edges between the structural root and spatially close leaves, this operation ensures β1(Γϵ)>0 _1( _ε)>0. It maps the 3D proximity of branches into the fundamental homology groups H1(Γϵ,ℤ)H_1( _ε,Z) of the graph, enabling the projection of the graph into a high-dimensional torus. Quotient Space Construction The augmented graph Γϵ _ε may still contain short terminal branches caused by data acquisition errors (cut-edges disjoint from ker(∂1) ( _1)). To remove these branches without altering the cycle basis, we contract them using a quotient space. Definition 0.1 (Quotient Metric Space). Let EbridgeE_bridge be the set of cut-edges in Γϵ _ε. Given a length threshold τ>0τ>0, define E<τ=e∈Ebridge∣ℓ(e)<τE_<τ=\e∈ E_bridge (e)<τ\. We define an equivalence relation ∼τ _τ on VcoreV_core such that u∼τvu _τv if and only if there exists a path between u and v consisting exclusively of edges in E<τE_<τ. The quotient space Γ~ϵ=Γϵ/∼τ _ε= _ε/ _τ is obtained by identifying all vertices within each equivalence class into a single vertex. To implement this operation algebraically, we define a weight matrix W. For any edge in E<τE_<τ, setting Wu,v→∞W_u,v→∞ in the discrete Laplace-Beltrami operator forces the resulting metric distance between u and v to zero. In implementation, substituting ∞ with a large scalar (e.g., 10910^9) and computing the Moore-Penrose pseudoinverse executes the vertex contraction deterministically. Figure 2: Distance evaluation on covering spaces. (e) Explicit lattice approximations compute the tropical polarization distance by solving the Closest Vector Problem (CVP) on the integer lattice Λ . Babai’s rounding maps the continuous target to a discrete lattice point (square), inducing a quantization error δ (red dashed vector). (f) Our formulation evaluates the exact unquantized minimum on the universal cover ℝgR^g using the Moore-Penrose pseudoinverse of the discrete Laplace-Beltrami operator (L+L^+), without requiring integer lattice searches. (g) Projection onto the Riemannian torus Jac(Γ)=ℝg/ΛJac( )=R^g/ . The explicit CVP approximation introduces metric distortion δ, whereas the discrete Arakelov-Green measure evaluates the continuous distance without quantization error. Tropical Jacobian and the Role of Polarization Distance After augmentation and quotienting, the graph Γ~ϵ _ε contains g=β1(Γ~ϵ)g= _1( _ε) fundamental cycles. Through the Abel-Jacobi map, this graph is projected into a g-dimensional torus, the Tropical Jacobian Jac(Γ~ϵ)=ℝg/ΛJac( _ε)=R^g/ , where Λ is the lattice generated by the fundamental cycles. A torus ℝg/ΛR^g/ is a quotient space. To measure the distance between two points on this torus, an inner product on the universal cover ℝgR^g is required. The tropical polarization provides this specific metric tensor, defined by the period matrix Q∈ℝg×gQ ^g× g of the graph. Assigning the polarization distance to the torus has an advantage: rather than relying on a single shortest path between two nodes, the polarization distance integrates spatial information across the entire cycle basis. This allows the metric to encode the global spatial configuration of the graph. The Lattice Approximation Baseline and Quantization Error Figure 2 contrasts this explicit lattice construction (panels e and g) with the continuous evaluation developed in the following subsection (panel f). Computing the exact polarized distance between two points x and y on the discrete torus ℝg/ΛR^g/ requires finding the optimal lattice translation n∈ℤgn ^g that minimizes the distance. This is mathematically equivalent to the Closest Vector Problem (CVP) on the lattice Λ : dTrop(x,y)=minn∈ℤg((x−y−Qn)⊤Q−1(x−y−Qn))12d_Trop(x,y)= _n ^g ((x-y-Qn) Q^-1(x-y-Qn) ) 12 (1) Solving the CVP is NP-Hard. Practical implementations must rely on polynomial-time approximations, such as Babai’s rounding algorithm. Let t denote the continuous coordinate difference before applying the lattice constraints. Babai’s algorithm maps t to a discrete lattice point by rounding: n^=⌊Q−1t⌉ n= Q^-1t . This approximation introduces a quantization error vector δ=Q−1t−n^δ=Q^-1t- n. Consequently, the approximated squared distance deviates from the true minimum by a discrepancy proportional to the quadratic form δTQδ^TQδ. As demonstrated in our experiments (Section Continuous Evaluation vs. Explicit Lattice Approximation), this quantization error distorts the structural representation, leading to performance degradation in graph classification tasks. This motivates evaluating the continuous relaxation on the universal cover instead, which requires no integer search and, as Table 3 shows, also yields a more discriminative descriptor. Closed-Form Unquantized Evaluation on the Universal Cover We compute pairwise distances on the quotient space Γ~ϵ _ε using discrete potential theory. Let L=diag(W)−WL=diag(W1)-W be the graph Laplacian. We compute its Moore-Penrose pseudoinverse L+L^+. The Arakelov-Green measure MAG∈ℝ|Vcore|×|Vcore|M_AG ^|V_core|×|V_core| is evaluated via the quadratic form: MAG(x,y)=Lx,x++Ly,y+−2Lx,y+M_AG(x,y)=L^+_x,x+L^+_y,y-2L^+_x,y (2) Evaluating exact distances on the Tropical Jacobian Jac(Γ~ϵ)=ℝg/ΛJac( _ε)=R^g/ requires solving the CVP on the integer lattice Λ , which is NP-Hard (Cao and Monod 2025). Instead of computing this discrete integer projection, we prove that MAGM_AG evaluates the continuous minimum of the Dirichlet energy. Mathematically, MAGM_AG incorporates the unquantized polarization distance on the universal cover ℝgR^g as an exact subtractive correction to the intrinsic path metric: MAG(x,y)=‖pxy‖W−12−(ΔΦxy)⊤Q−1(ΔΦxy)M_AG(x,y)=\|p_xy\|_W^-1^2-( _xy) Q^-1( _xy) (3) where ‖pxy‖W−12\|p_xy\|_W^-1^2 is the base path metric, and the second term is the squared unquantized distance on the universal cover ℝgR^g. This continuous formulation naturally encodes the geometric polarization induced by spatial cycles, and is obtained in closed form without any integer lattice search. Section Continuous Evaluation vs. Explicit Lattice Approximation shows that this continuous relaxation also avoids the quantization error of explicit rounding, yielding consistently higher accuracy than the explicit baseline. Spectral Feature Extraction Because the generalized inverse L+L^+ is positive semi-definite, there exists a matrix B such that L+=BBTL^+=B^T. The quadratic form can be rewritten as MAG(x,y)=‖bx−by‖22M_AG(x,y)=\|b_x-b_y\|_2^2, showing that MAGM_AG is a conditionally negative definite matrix that induces a Euclidean Distance Matrix (EDM). To derive a fixed-size representation, we compute its eigenvalue spectrum Σ=λ1,λ2,…,λ|Vcore| =\ _1, _2,…, _|V_core|\. For any permutation matrix P, the permuted matrix PMAGPTPM_AGP^T yields the same spectrum. Therefore, we extract the top-K absolute eigenvalues Γ=TopK(SortDescending(|Σ|),K)S_ =TopK(SortDescending(| |),K) as a permutation-invariant signature. This structural prior is then concatenated with node or graph features in neural architectures. Algorithm 1 Spectral Signature via Arakelov-Green Measure Input: Spatial tree =(V,E,ι)T=(V,E, ), proximity threshold ϵε, contraction threshold τ, feature dimension K. Output: Permutation-invariant spectral signature Γ∈ℝKS_ ^K. 1: Step 1: Topological Retraction 2: Γmin←KronReduction(LCC) _ (T_LCC) 3: Step 2: Cycle Space Augmentation 4: Eϵ←(v,vroot)∣v∈Vcore, leaf,‖ι(v)−ι(vroot)‖2<ϵE_ε←\(v,v_root) v∈ V_core, leaf,\| (v)- (v_root)\|_2<ε\ 5: Γϵ←Γmin∪Eϵ _ε← _ ∪ E_ε 6: Step 3: Quotient Space Construction 7: Ebridge←CutEdges(Γϵ)E_bridge ( _ε) 8: Construct weight matrix W∈ℝ|Vcore|×|Vcore|W ^|V_core|×|V_core| where: 9: Wu,v=∞,if eu,v∈Ebridge and ℓ(eu,v)<τ1/ℓ(eu,v),otherwiseW_u,v= cases∞,&if e_u,v∈ E_bridge and (e_u,v)<τ\\ 1/ (e_u,v),&otherwise cases 10: Step 4: Arakelov-Green Measure Evaluation 11: L←diag(W)−WL (W1)-W 12: L+←GeneralizedInverse(L)L^+ (L) 13: MAG←diag(L+)T+diag(L+)T−2L+M_AG (L^+)1^T+1diag(L^+)^T-2L^+ 14: Step 5: Spectral Extraction 15: Σ←Spectrum(MAG) (M_AG) 16: Γ←TopK(SortDescending(|Σ|),K)S_ (SortDescending(| |),K) 17: return ΓS_ Experiments We empirically evaluate our framework’s expressiveness against the 1-Weisfeiler-Lehman (1-WL) test on graph isomorphism benchmarks, and assess its effectiveness as a structural prior on 3D neuronal morphologies. Theoretical Validation: Graph Isomorphism Standard message-passing networks are 1-WL bounded, limiting their ability to distinguish certain spatial cycles. We evaluate our metric’s expressive power on the BREC dataset (Wang and Zhang 2024), comprising 400 symmetric graph pairs. Model / Type Basic Regular Extend CFI Total Theoretical (60) (140) (100) (100) (400) 3-WL Test 100% 35.7% 100% 60.0% 67.5% Transformers Graphormer 26.7% 8.6% 41.0% 10.0% 19.8% High-Order GNNs NGNN (Subgraph) 98.3% 34.3% 59.0% 0.0% 41.5% PPGN (k-WL) 100% 35.7% 100% 23.0% 58.2% KP-GNN (Subgraph) 100% 75.7% 98.0% 11.0% 68.8% I2-GNN (Subgraph) 100% 71.4% 100% 21.0% 70.2% AG-GIN (Ours) 100% 62.1% 100% 33.0% 70.0% Table 1: Pair distinguishing accuracies on the BREC benchmark. The AG-GIN demonstrates expressivity beyond standard 1-WL message passing. Setup & Analysis: After computing the discrete Arakelov-Green distance matrix, we extract its absolute eigenvectors to ensure permutation equivariance and resolve sign ambiguity. These structural coordinates are concatenated with standard node features and processed via a Graph Isomorphism Network augmented with Jumping Knowledge (AG-GIN). As detailed in Table 1, AG-GIN separates 70.0%70.0\% of total pairs, achieving 100%100\% on Basic and Extension categories. It yields 33.0%33.0\% on the challenging CFI category, outperforming higher-order/subgraph GNNs like I2-GNN (21.0%21.0\%) and PPGN (23.0%23.0\%). This confirms our metric maps non-isomorphic cyclic structures to distinct representations. Furthermore, unlike subgraph GNNs requiring (|V|3)O(|V|^3) enumerations per training step, our operator evaluates the metric algebraically in a single preprocessing step. Empirical Evaluation on 3D Morphologies Datasets: We evaluate on three biological datasets configured following prior benchmarks (Chen et al. 2022): ACT (Allen Cell Types) (Gouwens et al. 2019) with 495 manually labeled neurons from the mouse visual cortex (reconstructions are flat due to thin slicing); JML (Janelia MouseLight) (Gao et al. 2023; Winnubst et al. 2019) comprising 505 automatically labeled, complete projection neurons; and BIL (BICCN fMOST) (Peng et al. 2021) containing 1,413 manually labeled, complete neurons from diverse regions (cortex, claustrum, striatum, thalamus). Evaluation Protocol & Hyperparameters: We employ 10-fold cross-validation with absolute coordinate normalization, using folds 0–7 for training and 8–9 for a strict single-evaluation test set. Results report the mean and standard deviation across 5 random seeds at the best-validation epoch. Models are optimized via AdamW (peak learning rate 10−310^-3 or 10−410^-4, weight decay 10−410^-4) with cosine annealing (Tmax=150T_ =150, ηmin=10−6 _ =10^-6) and gradient clipping at 1.0. Training terminates with a patience of 25 epochs, utilizing batch sizes of 128 (tree-based) or 64 (point-cloud). Method / Architecture ACT-4 JML-4 BIL-6 Sequence and Tree-Based Paradigms TRNN (Translation-Inv.) 33.70±1.4533.70± 1.45 42.90±1.6242.90± 1.62 31.00±1.8031.00± 1.80 TreeMoCo (Chen et al. 2022) (k-N, Unsupervised) 60.21±1.7360.21± 1.73 63.16±0.9363.16± 0.93 77.36±0.8977.36± 0.89 TreeMoCo (Chen et al. 2022) (Fine-tuned, Supervised) 57.05±1.7357.05± 1.73 72.89±1.5072.89± 1.50 87.19±1.1387.19± 1.13 GraPHFormer (Tree Branch) (Shah et al. 2026) 54.52±1.3754.52± 1.37 73.33±2.6773.33± 2.67 82.97±0.5482.97± 0.54 TreeLSTM (+ AG Prior) [Ours] 69.68±3.6169.68± 3.61 74.29±2.2374.29± 2.23 93.55±0.5693.55± 0.56 Graph-Based Paradigms MorphoGNN (Zhu et al. 2023) 52.21±2.2752.21± 2.27 65.19±2.5265.19± 2.52 88.45±0.7588.45± 0.75 Ours (GNN Integration) 54.32±2.7154.32± 2.71 (+2.11) 76.36±0.9776.36± 0.97 (+11.17) 92.15±0.4592.15± 0.45 (+3.70) Generative Paradigms MorphVAE (Laturnus and Berens 2021) 44.63±1.7644.63± 1.76 51.95±5.1951.95± 5.19 55.04±1.9755.04± 1.97 MorphVAE (+ AG Prior) 48.84±1.6048.84± 1.60 (+4.21) 61.56±4.0761.56± 4.07 (+9.61) 75.29±0.6175.29± 0.61 (+20.25) Specialized Spatial Transformers (Different Preprocessing) SGTMorph (Supervised) (Sheng et al. 2025) – 72.4072.40 88.9088.90 Table 2: Classification accuracy (%) on 3D morphological benchmarks. We partition the models by their architectural paradigms. For Graph and Generative paradigms, we report controlled ablations against their respective baselines to isolate the exact gain (Δ ) of our structural prior. All results are from our own reimplementation under an identical pipeline, split, and seeds, except for SGTMorph, whose results are quoted from the literature due to specific preprocessing dependencies. Necessity of Spatial Coordinates Under translation-invariant settings, where absolute Cartesian anchors are removed to prevent coordinate memorization, TRNN (a topological sequence model) degrades severely (33.70%33.70\% on ACT-4, 31.00%31.00\% on BIL-6). This confirms sequential topological traversals lose spatial awareness without absolute coordinate mappings. Analysis of Transformer-based Architectures Recent Transformer architectures utilize specialized preprocessing pipelines complicating direct structural comparisons. To ensure controlled evaluations, we analyze their components mechanically. GraPHFormer (Shah et al. 2026) relies on multi-modal continuous vectorization (2D persistence images via Gaussian kernel density estimation) altering the original graph metric. To isolate its graph representation capability, we evaluate its native TreeLSTM-Double branch under our 80%–20% split, yielding 54.52%54.52\% (ACT-4), 73.33%73.33\% (JML-4), and 82.97%82.97\% (BIL-6). Our TreeLSTM augmented with the AG prior outperforms this branch across all datasets, isolating the full GraPHFormer’s remaining performance gap to its multi-modal image augmentations. See appendix for details. SGTMorph: The SGTMorph architecture (Sheng et al. 2025) is tightly coupled with a specialized preprocessing pipeline involving pre-computed heuristics, min-max box-normalization, and random jitter. Forcing this pipeline into our strict absolute coordinate protocol would fundamentally alter their intended data distribution, risking an unfair comparison. Instead, to rigorously isolate their morphological learning capacity, we independently constructed and evaluated a positional probe utilizing their exact min-max normalization strategy. Predicting labels purely from these normalized soma coordinates yields 61.0%61.0\% on JML-4 and 74.8%74.8\% on BIL-6. Crucially, our empirical probe results remain significantly lower than the full-model accuracies published in their paper (72.4%72.4\% and 88.9%88.9\%, respectively). This validates that SGTMorph effectively captures true morphological structures rather than merely exploiting coordinate leakage. Ultimately, while their method requires transforming the spatial input, our Arakelov-Green framework operates directly on unaugmented metric graphs, yielding superior structural discrimination (e.g., 93.55%93.55\% vs. their 88.90%88.90\% on BIL-6). See appendix for details. Integration Protocols for Baseline Architectures Let ∈ℝdss ^d_s denote the pre-computed Arakelov-Green (AG) spectral signature for a given input graph G (ds=64d_s=64). Let MLPspec:ℝds→ℝdpMLP_spec:R^d_s ^d_p be a multi-layer perceptron with Layer Normalization mapping the spectral signature to a dpd_p-dimensional representation. The operator ⊕ denotes feature-wise concatenation. 1. Integration in Tree-based Paradigms. We evaluate our TreeLSTM model against TreeMoCo (Chen et al. 2022), which utilizes a Momentum Contrast objective. For TreeMoCo, we report both its unsupervised k-N performance (using 334,720 trainable parameters for the backbone) and its supervised fine-tuned performance (using up to 335,494 trainable parameters). To represent our approach in this paradigm, we use a supervised TreeLSTM. Under the default configuration (hidden dimension 128), our TreeLSTM combined with the AG prior projection contains only 97,792 trainable parameters. Let root∈ℝdhh_root ^d_h be the final hidden state aggregated at the root node. The spectral feature is integrated via late fusion: Tree_fused=MLPgraph(root)⊕MLPspec()∈ℝdh′+dph_Tree\_fused=MLP_graph(h_root) _spec(s) ^d_h +d_p (4) As shown in Table 2, despite utilizing fewer than one-third of the parameters, our TreeLSTM (+ AG Prior) outperforms the fine-tuned TreeMoCo on all datasets (e.g., 93.55%93.55\% vs 87.19%87.19\% on BIL-6). 2. Integration in MorphoGNN. The model samples the neuron into a point cloud and processes it via EdgeConv layers. Let pc∈ℝ2048v_pc ^2048 denote the global point cloud representation. The spectral feature is injected post-pooling: Morpho_fused=pc⊕MLPspec()h_Morpho\_fused=v_pc _spec(s). 3. Integration in MorphVAE. A Seq2Seq framework encodes 3D random walks sampled from the neuron. Let rT∈ℝkr_T ^k represent the global latent vector. The spectral feature is appended to this latent vector: fused=rT⊕MLPspec()r_fused=r_T _spec(s). The projection layer (MLPspecMLP_spec) aligns the dimension and scale of the spectral features with the latent space of the backbone architectures. Training a standalone classifier exclusively on the spectral signature s without the neural backbone yields sub-optimal accuracy. This indicates that the performance improvements derive from the combination of the structural prior with the local operations of the baseline networks, rather than from the parameters introduced by the projection layer. Continuous Evaluation vs. Explicit Lattice Approximation Exact CVP evaluation on the Tropical Jacobian requires (2g)O(2^g) enumerations (g=β1(Γ~ϵ)g= _1( _ε)). At ϵ=50.0ε=50.0, maximum g reaches 106 (ACT-4), 35 (JML-4), and 30 (BIL-6), CVP limits (e.g., (2106)O(2^106)) computationally intractable and necessitating approximations like Babai’s rounding. To ensure strict comparison, we construct a controlled baseline using BFS fundamental cycles, computing period matrix Q, and approximating distances via Babai’s method. Using a frozen k-N classifier at ϵ=50.0,τ=10.0ε=50.0,τ=10.0, we compare this explicit lattice approximation against our continuous formulation (Table 3). Dataset CVP Baseline Continuous AG (Ours) Acc Mac-F1 Time(s) Acc Mac-F1 Time(s) ACT-4 47.37 45.20 109.2 49.47 47.79 16.8 JML-4 49.35 32.49 376.7 48.05 36.21 103.0 BIL-6 52.89 41.27 908.1 64.05 46.65 219.1 Table 3: Performance and efficiency comparison between the explicit lattice approximation (CVP) and the continuous evaluation (AG) at ϵ=50.0,τ=10.0ε=50.0,τ=10.0. Time denotes the preprocessing cost in seconds. The continuous formulation demonstrates specific advantages: 1. Avoidance of Quantization Error and Basis Ambiguity: The rounding operation n^=⌊Q−1t⌉ n= Q^-1t introduces quantization errors and does not commute with basis transformation matrices U∈GLg(ℤ)U∈ GL_g(Z), making the explicit descriptor basis-dependent. Conversely, our continuous formulation evaluates deterministically without basis extraction, avoiding quantization errors and consistently yielding higher Macro F1 scores (though CVP retains a marginal simple accuracy edge on the imbalanced JML-4). 2. Algorithmic Efficiency: Continuous evaluation reduces preprocessing time by 4.1×4.1×–6.5×6.5× via fully parallelizable dense linear algebra (a single matrix pseudoinverse). Explicit CVP suffers from sequential graph traversals for homology extraction, memory inflation by a factor of g (projecting N nodes into ℝg×N×NR^g× N× N), and dynamic programming bottlenecks (e.g., Floyd-Warshall (|V|3)O(|V|^3)) to compute all-pairs shortest paths for acyclic backbone metric compensation. Ablation Study To isolate the contributions of algebraic operations, we ablate the spatial proximity threshold ϵε (cycle space augmentation) and structural threshold τ (quotient space vertex contraction) using a frozen k-N classifier to remove neural network optimization confounders (Table 4). ϵε τ ACT-4 JML-4 BIL-6 Acc Mac-F1 Acc Mac-F1 Acc Mac-F1 0.0 0.0 41.05 38.71 49.35 32.48 52.07 38.87 50.0 0.0 45.26 43.09 49.35 32.48 52.48 40.12 0.0 10.0 46.32 44.73 45.45 39.69 63.64 47.24 50.0 10.0 49.47 47.79 48.05 36.21 64.05 46.65 Table 4: Ablation Study of the Proposed Method. Performance is measured in Accuracy (%) and Macro F1 (%). Effects of Cycle and Quotient Spaces: The control setting (ϵ=0.0,τ=0.0ε=0.0,τ=0.0) algebraically reduces to standard resistance distance. Inducing spatial 1-simplices (ϵ>0ε>0) encodes spatial proximities into homological cycles, while edge contraction (τ>0τ>0) forms a quotient space contracting terminal variations. Combined operations (ϵ=50.0,τ=10.0ε=50.0,τ=10.0) yield the highest structural discrimination on ACT-4 and BIL-6. On JML-4, comprising complete and uniform projection neurons, terminal branches represent genuine morphology rather than acquisition artifacts. Consequently, their contraction is sub-optimal for absolute accuracy (though it improves Macro F1 on imbalanced classes), and the two operations do not constructively compose. Conclusion The quantitative representation of 3D neuronal morphologies requires capturing both graph topology and spatial geometry. In this work, we address this requirement using tropical algebraic geometry. Evaluating exact distances on the discrete Tropical Jacobian requires solving the NP-Hard Closest Vector Problem (CVP) on integer lattices. Instead of computing this discrete integer projection, we adopt a continuous relaxation on the universal cover ℝgR^g. We prove that the discrete Arakelov-Green measure, computed in closed form via the generalized inverse of the discrete Laplace-Beltrami operator, decomposes exactly into the intrinsic path metric minus the unquantized polarization distance on this continuous space, and is therefore obtained without any integer lattice search. This approach avoids the quantization errors associated with discrete lattice approximations, yielding superior empirical accuracy. This metric yields two distinct structural descriptors. Its eigenvector formulation provides node-level coordinates that demonstrate expressivity beyond the standard 1-WL test on the BREC benchmark. Its permutation-invariant eigenvalue spectrum provides a graph-level signature that improves the classification accuracy of standard architectures (MLPs, GNNs, Tree-LSTMs) on 3D morphology datasets without additional trainable parameters. Overall, it offers an efficient, training-free geometric prior for graph representation learning. Limitations and Future Work Our framework provides a deterministic structural prior with (|Vcore|3)O(|V_core|^3) preprocessing complexity. A core design choice of this method is the deliberate removal of absolute coordinate anchoring. Parameterized spatial models relying on dense coordinate regression may achieve high accuracy on specific datasets by memorizing absolute spatial distributions. In contrast, our signature isolates geometric and topological invariants. While this discards dataset-specific positional cues, it prevents coordinate memorization and ensures invariance to continuous spatial transformations. This trade-off presents a distinct direction for future research. The mathematical decomposition of the discrete Arakelov-Green measure over the continuous universal cover opens new avenues for graph network design. 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Let Γ=(V,E,ℓ) =(V,E, ) be a connected metric graph with symmetric affinity matrix W∈ℝ|V|×|V|W ^|V|×|V|, where Wu,v=1/ℓ(u,v)W_u,v=1/ (u,v) for (u,v)∈E(u,v)∈ E. The discrete Laplace-Beltrami operator is L=diag(W)−WL=diag(W1)-W. Partition the vertex set as V=Vcore∪VregV=V_core∪ V_reg, where VregV_reg contains exclusively degree-2 vertices. The Schur complement of L with respect to VregV_reg is LS=Lcore−LmixLreg−1Lmix⊤L_S=L_core-L_mixL_reg^-1L_mix . Theorem A.1 (Metric Preservation). For any vertices x,y∈Vcorex,y∈ V_core, the discrete metric evaluated by the generalized inverse L+L^+ on the original graph is identical to the metric evaluated by LS+L_S^+ on the reduced graph. Furthermore, the reciprocal of the updated non-zero off-diagonal entries in LSL_S equals the Lebesgue sum of the intrinsic metric lengths along the retracted paths. Proof. The Laplacian quadratic form ⊤Lv Lv quantifies the Dirichlet energy of a discrete potential function v. Minimizing this quadratic form over the intermediate variables VregV_reg subject to fixed boundary conditions on VcoreV_core defines the harmonic extension, which is resolved by the Schur complement LSL_S. Because the discrete metric between x,y∈Vcorex,y∈ V_core is the inverse of the minimal quadratic form required to establish a unit potential difference, and since no external constraints are placed on VregV_reg, the internal Gaussian elimination preserves the global minimum. Thus, Lxx++Lyy+−2Lxy+=(LS+)xx+(LS+)yy−2(LS+)xyL^+_x+L^+_y-2L^+_xy=(L_S^+)_x+(L_S^+)_y-2(L_S^+)_xy. For an isolated degree-2 node v∈Vregv∈ V_reg connected to u,w∈Vcoreu,w∈ V_core, the block LregL_reg reduces to a scalar Wuv+WvwW_uv+W_vw. The Schur elimination updates the off-diagonal affinity between u and w as: −(LS)uw=Wuw′=Wuw+WuvWvwWuv+Wvw-(L_S)_uw=W _uw=W_uw+ W_uvW_vwW_uv+W_vw Assuming Wuw=0W_uw=0 (no initial edge), the new consolidated affinity is Wuw′=WuvWvwWuv+WvwW _uw= W_uvW_vwW_uv+W_vw. Taking the algebraic reciprocal yields the intrinsic metric length: 1Wuw′=Wuv+WvwWuvWvw=1Wuv+1Wvw=ℓ(u,v)+ℓ(v,w) 1W _uw= W_uv+W_vwW_uvW_vw= 1W_uv+ 1W_vw= (u,v)+ (v,w) This equates to the sum of intrinsic segment lengths forming the retracted macroscopic path. ∎ A.2 Betti Number and Homological Augmentation Definition A.2. Let =(V,E,ι)T=(V,E, ) be a spatial tree, characterized by a first Betti number β1()=0 _1(T)=0. Let Eϵ=e1,e2,…,ekE_ε=\e_1,e_2,…,e_k\ be k adjoined 1-simplices forming the augmented spatial graph Γϵ=∪Eϵ _ε=T∪ E_ε. Theorem A.2 (Homological Augmentation). The augmented graph Γϵ _ε possesses a first Betti number β1(Γϵ)=k _1( _ε)=k. Additionally, each augmented 1-simplex ei∈Eϵe_i∈ E_ε maps bijectively to a unique, linearly independent 1-cycle generator in ker(∂1) ( _1). Proof. We invoke the Euler-Poincaré formula for a connected topological space, χ=|V|−|Etotal|=β0−β1χ=|V|-|E_total|= _0- _1. For the baseline tree T, since β0=1 _0=1 and β1=0 _1=0, we have |V|−|E|=1|V|-|E|=1. For the augmented graph Γϵ _ε, the total edge count is |E|+k|E|+k. Because edge adjunction onto a connected component preserves connectedness, β0 _0 remains 1. The Euler characteristic of Γϵ _ε evaluates to: |V|−(|E|+k)=(|V|−|E|)−k=1−k|V|-(|E|+k)=(|V|-|E|)-k=1-k Equating this to β0−β1=1−β1 _0- _1=1- _1 yields 1−k=1−β1⟹β1=k1-k=1- _1 _1=k. To construct the explicit homology basis, isolate any augmented edge ei=(ui,vi)∈Eϵe_i=(u_i,v_i)∈ E_ε. Because T is a tree, there exists a simple path pi⊂Ep_i⊂ E connecting viv_i back to uiu_i. The formal chain sum zi=ei+piz_i=e_i+p_i is closed (∂1zi=0 _1z_i=0), forming a 1-cycle. The collection z1,z2,…,zk\z_1,z_2,…,z_k\ is linearly independent because each ziz_i contains exactly one simplex ei∉supp(zj)e_i (z_j) for all j≠ij≠ i. Given dim(ker(∂1))=β1=k ( ( _1))= _1=k, this set forms a complete and bijective basis for the fundamental homology group H1(Γϵ,ℤ)H_1( _ε,Z). ∎ A.3 Metric Quotient Space via Affinity Limits Theorem A.3 (Metric Quotient Isometry). Let euv=(u,v)e_uv=(u,v) be a cut-edge in Γϵ _ε, such that it belongs to the support of no cycle z∈ker(∂1)z∈ ( _1). The analytic limit limWu,v→∞MAG(u,v)=0 _W_u,v→∞M_AG(u,v)=0 guarantees a well-defined equivalence relation ∼τ _τ, inducing a quotient metric space Γ~ϵ=Γϵ/∼τ _ε= _ε/ _τ wherein the topology of the macroscopic cycle space remains invariant. Proof. By discrete potential theory, evaluating MAG(u,v)=Luu++Lvv+−2Luv+M_AG(u,v)=L^+_u+L^+_v-2L^+_uv equates to solving for the minimal W−1W^-1-norm of a 1-chain flow fuvf_uv satisfying the boundary condition ∂1fuv=δu−δv _1f_uv= _u- _v. Because euve_uv is a cut-edge, any valid 1-chain satisfying this boundary condition must assign a coefficient of 1 to euve_uv, as its removal disconnects the graph. The minimal quadratic form ‖fuv‖W−12\|f_uv\|^2_W^-1 is dictated by the inverse affinity of the edge itself, yielding MAG(u,v)=1/Wu,vM_AG(u,v)=1/W_u,v. Taking the limit Wu,v→∞W_u,v→∞, we obtain MAG(u,v)→0M_AG(u,v)→ 0. Since this quadratic form satisfies metric axioms on any connected graph, coercing the distance between adjacent nodes to zero transforms the metric into a pseudometric. Defining x∼τy⇔MAG(x,y)=0x _τy M_AG(x,y)=0 constitutes a valid equivalence relation, collapsing the bridge euve_uv into a singular quotient vertex. Because euve_uv is disjoint from ker(∂1) ( _1), its contraction alters neither the cycle basis nor the Betti number β1 _1. ∎ A.4 Continuous Identity on the Universal Cover Theorem A.4 (Continuous Identity). Let fxy∈C1(Γ~ϵ;ℝ)f_xy∈ C_1( _ε;R) be a 1-chain satisfying ∂1fxy=δx−δy _1f_xy= _x- _y. Let Z1=ker(∂1)Z_1= ( _1) be defined by the cycle basis C∈ℝg×|E|C ^g×|E|, with the period matrix Q=CW−1C⊤Q=CW^-1C , and the continuous projection map ΔΦxy=CW−1pxy _xy=CW^-1p_xy. The minimization of the 1-chain path norm yields MAG(x,y)=‖pxy‖W−12−(ΔΦxy)⊤Q−1(ΔΦxy)M_AG(x,y)=\|p_xy\|_W^-1^2-( _xy) Q^-1( _xy), where the latter subtractive term represents the squared distance on the universal cover ℝgR^g of the Albanese torus. Proof. By the discrete Hodge orthogonal decomposition, any 1-chain satisfying the boundary conditions can be parameterized as fxy=pxy+C⊤cf_xy=p_xy+C c for a continuous cyclic coefficient vector c∈ℝgc ^g, where pxyp_xy is a base 1-chain. We minimize the quadratic functional: ℰ(c) (c) =‖pxy+C⊤c‖W−12 =\|p_xy+C c\|_W^-1^2 =‖pxy‖W−12+2c⊤ΔΦxy+c⊤Qc =\|p_xy\|_W^-1^2+2c _xy+c Qc Because Q is a positive-definite Gram matrix, the functional is convex. Setting the gradient with respect to c to zero yields the unique global minimum c∗=−Q−1ΔΦxyc^*=-Q^-1 _xy. Substituting c∗c^* back into the functional gives: MAG(x,y) M_AG(x,y) =ℰ(c∗) =E(c^*) =‖pxy‖W−12−2(ΔΦxy)⊤Q−1(ΔΦxy) =\|p_xy\|_W^-1^2-2( _xy) Q^-1( _xy) +(ΔΦxy)⊤Q−1QQ−1(ΔΦxy) +( _xy) Q^-1Q^-1( _xy) =‖pxy‖W−12−(ΔΦxy)⊤Q−1(ΔΦxy) =\|p_xy\|_W^-1^2-( _xy) Q^-1( _xy) The subtractive term evaluates the squared metric distance of the unquantized projection ΔΦxy _xy mapped onto the continuous universal cover ℝgR^g. ∎ A.5 Geometric Embeddability and Permutation Invariance Theorem A.5 (Embeddability and Spectral Invariance). (1) MAGM_AG operates as a conditionally negative definite (CND) Euclidean Distance Matrix (EDM). (2) For any vertex permutation matrix P∈0,1N×NP∈\0,1\^N× N, the spectra (multisets of eigenvalues) of MAGM_AG and PMAGP⊤PM_AGP are identical. Proof. (1) Express MAG=d⊤+d⊤−2L+M_AG=d1 +1d -2L^+, where d=diag(L+)d=diag(L^+). To prove CND, we evaluate the quadratic form for an arbitrary vector v satisfying ⊤=01 v=0: ⊤MAG=⊤(d⊤+d⊤−2L+)v M_AGv=v (d1 +1d -2L^+)v Since ⊤=0⟹⊤=01 v=0 1=0, the rank-1 outer products vanish: ⊤(d⊤)=0v (d1 )v=0 and ⊤(d⊤)=0v (1d )v=0. This leaves ⊤MAG=−2⊤L+v M_AGv=-2v L^+v. Because L+L^+ is positive semi-definite, ⊤L+≥0v L^+v≥ 0. Consequently, ⊤MAG≤0v M_AGv≤ 0, proving it conditionally negative definite and isomorphic to a valid EDM mapping. (2) A permutation matrix P is orthogonal (P⊤P=IP P=I). The conjugated Laplacian is L~=PLP⊤ L=PLP , yielding the generalized inverse L~+=PL+P⊤ L^+=PL^+P and diagonal d~=Pd d=Pd. Recognizing the permutation invariance of the all-ones vector (⊤=⊤P⊤1 =1 P ), the permuted metric matrix evaluates as: M~AG=P(d⊤+d⊤−2L+)P⊤=PMAGP⊤ M_AG=P (d1 +1d -2L^+ )P =PM_AGP Because M~AG M_AG is algebraically similar to MAGM_AG under the orthogonal matrix P, their eigenvalue spectra are identical. ∎ A.6 Continuous Universal Cover vs. Explicit Lattice Approximation Theorem A.6 (Continuous Minimum vs. Lattice Discrepancy). The algebraic formulation via L+L^+ evaluates the continuous minimum of the energy functional on the universal cover, avoiding the integer lattice quantization errors introduced by explicit CVP approximations. Proof. Evaluating explicit distances on the Tropical Jacobian requires solving the Closest Vector Problem (CVP) on the lattice Λ=Qℤg =QZ^g. Let t=ΔΦxyt= _xy denote the unquantized continuous projection. Explicit combinatorial solutions (e.g., Babai’s rounding) yield an approximate lattice vector n^∈ℤg n ^g mapped to Λ , incurring a quantization error δ=Q−1t−n^∈[−0.5,0.5]gδ=Q^-1t- n∈[-0.5,0.5]^g. The explicitly approximated lattice distance takes the form dBabai2=(t−Qn^)⊤Q−1(t−Qn^)d_Babai^2=(t-Q n) Q^-1(t-Q n). Substituting the approximation error t−Qn^=Qδt-Q n=Qδ yields: dBabai2=(Qδ)⊤Q−1(Qδ)=δ⊤Qδd_Babai^2=(Qδ) Q^-1(Qδ)=δ Qδ Constructive explicit approaches incur a discrepancy proportional to the quadratic form δ⊤Qδ Qδ, which is dependent on the period matrix Q and non-zero for a generic continuous vector. Conversely, based on Theorem A.4, the discrete Arakelov-Green metric evaluates the continuous minimum of the energy functional across the unquantized domain c∈ℝgc ^g: MAG(x,y)=‖pxy‖W−12−t⊤Q−1tM_AG(x,y)=\|p_xy\|_W^-1^2-t Q^-1t Rather than mapping t onto the discrete lattice Λ , this formulation computes the unquantized squared distance t⊤Q−1t Q^-1t on the universal cover ℝgR^g. By subtracting this continuous polarization term natively via L+L^+, our approach avoids the integer lattice constraints and removes the quantization error δ⊤Qδ Qδ. Remark A.1 (Basis-Invariance and Quantization Error). The derivation provides two mathematical insights that support continuous evaluation: (1) Quantization Error Check: If Babai’s rounding returns the zero vector (n^= n=0), then δ=Q−1tδ=Q^-1t, and the explicit lattice approximation yields dBabai2=t⊤Q−1td_Babai^2=t Q^-1t. In this scenario, the CVP term and the continuous subtractive term are mathematically identical. The performance gap between the explicit CVP baseline and the Arakelov-Green evaluation originates from non-zero lattice roundings (n^≠ n 0). (2) Basis-Invariance: Let U∈GLg(ℤ)U∈ GL_g(Z) represent a change of the fundamental cycle basis, mapping C↦UC UC. Under this transformation, Q↦UQU⊤Q UQU and t↦Ut Ut. The continuous subtractive term transforms as t⊤U⊤(UQU⊤)−1Ut=t⊤Q−1t U (UQU )^-1Ut=t Q^-1t. Thus, the continuous evaluation is basis-invariant. In contrast, Babai’s rounding operation n^=⌊Q−1t⌉ n= Q^-1t does not commute with U−1U^-1. As a result, the explicit CVP approximation yields different structural descriptors depending on the arbitrary choice of the spanning tree. ∎ A.7 Construction of the Explicit Baseline This section provides the mathematical derivation of the explicit CVP baseline approximation used in our experiments, whose quantization error is analyzed in Section A.6. Matrix Constructions and the CVP Formulation Let G=(V,E)G=(V,E) be the combinatorial model of the augmented metric graph Γ with |V|=n|V|=n and |E|=m|E|=m. We fix a fundamental basis of 1-cycles σi1≤i≤g\ _i\_1≤ i≤ g derived from a spanning tree. We define the cycle-edge incidence matrix C∈ℝg×mC ^g× m such that CijC_ij indicates the orientation of edge eje_j in cycle σi _i. Let Dℓ∈ℝm×mD_ ^m× m be the diagonal edge length matrix. The tropical period matrix Q∈ℝg×gQ ^g× g is constructed as Q=CDℓC⊤Q=CD_ C . For two coordinate vectors [x],[y][x],[y] embedded in the Tropical Jacobian Jac(Γ)Jac( ), the tropical polarization distance dTropd_Trop is tied to the CVP on the lattice generated by Q: dTrop([x],[y])=minn∈ℤg((x−y−Qn)⊤Q−1(x−y−Qn))12d_Trop([x],[y])= _n ^g ((x-y-Qn) Q^-1(x-y-Qn) ) 12 The CVP on general lattices is NP-hard (van Emde Boas 1981a). Finding the discrete minimum over n∈ℤgn ^g requires enumeration algorithms with an exponential time complexity (e.g., (2g)O(2^g)). Construction of the Explicit Approximation We now show how the orthogonal projection onto the harmonic space yields a polynomial-time approximation. Proof. Let Y∈ℝn×mY ^n× m be the path-edge incidence matrix, where YijY_ij indicates whether edge eje_j lies on the unique spanning-tree path from the root to node viv_i. The unquantized universal cover coordinates Vuniversal∈ℝg×nV_universal ^g× n are constructed as Vuniversal=CDℓY⊤V_universal=CD_ Y . Consider the orthogonal projection map π:C1(G;ℝ)→H1(G;ℝ)π:C_1(G;R)→ H_1(G;R). The matrix representing this projection with respect to the fundamental basis is Π=Q−1CDℓ =Q^-1CD_ . For any vertex i∈Vi∈ V, its path from the root is projected onto the continuous covering space as: V~:,i=ΠYi,:⊤=Q−1CDℓYi,:⊤=Q−1Vuniversal[:,i] V_:,i= Y_i,: =Q^-1CD_ Y_i,: =Q^-1V_universal[:,i] The term (V~:,i−V~:,j)( V_:,i- V_:,j) represents the continuous polarization difference. By applying Babai’s Rounding Algorithm, we approximate the discrete lattice shift n∗∈ℤgn^* ^g by taking the nearest integer vector of the continuous projection: n∗≈⌊V~:,i−V~:,j⌉n^*≈ V_:,i- V_:,j . Substituting this approximation yields the residual distance vector Δij _ij: Δij=(V~:,i−V~:,j)−⌊V~:,i−V~:,j⌉ _ij=( V_:,i- V_:,j)- V_:,i- V_:,j The quadratic form mapping back to the metric space is: DTorus2(i,j)=Δij⊤QΔijD_Torus^2(i,j)= _ij Q _ij This executes via dense matrix multiplications and element-wise rounding operations, limiting the time complexity to (|V|3)O(|V|^3) while bounding the continuous-to-discrete geometric distortion. ∎ Table 5: Dataset configurations and split statistics for the GraPHFormer reproduction. Dataset Source Classes Train (Balanced) Test ACT-4 Allen Cell Types Isocortex L2/3, L4, L5, L6 576 (144 × 4) 95 JML-4 Janelia MouseLight Isocortex L2/3, L5, L6, VPM 672 (168 × 4) 75 BIL-6 BICCN fMOST CP, VPM, Isocortex L2/3, L4, L5, L6 1824 (304 × 6) 242 Table 6: Performance comparison of the GraPHFormer reproduction against the published literature. Dataset Graph Branch (Ours, 80/20) Ref: Tree Encoder (70/30) Ref: Full Multimodal (70/30) ACT-4 54.52% ± 1.37% 53.5% 65.5% JML-4 73.33% ± 2.67% 76.5% 76.5% BIL-6 82.97% ± 0.54% 88.3% 93.51% Table 7: Positional probe evaluation for SGTMorph based on global min-max normalized soma coordinates. Dataset Our Positional Probe (Soma-only) Ref: Published Full Model (SL) JML-4 61.0% 72.4% BIL-6 74.8% 88.9% Table 8: Classification performance of the standalone MLP baseline utilizing the 64-dimensional Arakelov-Green spectral signature across 5 random seeds. Dataset Accuracy (Mean ± Std) Max Acc Macro-F1 (Mean ± Std) Params ACT-4 63.79% ± 1.20% 65.26% 62.67% ± 2.32% 11,092 JML-4 60.26% ± 1.72% 63.16% 60.00% ± 1.99% 11,092 BIL-6 73.31% ± 0.86% 74.38% 63.13% ± 2.43% 11,190 Table 9: Classification accuracy of the frozen MorphVAE baseline and the direct spectral concatenation model. Dataset MorphVAE Only (32D) Direct Concatenation (96D) Absolute Gain (Δ ) ACT-4 44.63% ± 1.76% 48.84% ± 1.60% +4.21% JML-4 51.95% ± 5.19% 61.56% ± 4.07% +9.61% BIL-6 55.04% ± 1.97% 75.29% ± 0.61% +20.25% A.8 Independent Reproduction of GraPHFormer This section details the independent reproduction of the graph (tree) branch of GraPHFormer (Shah et al. 2026) under our 80%/20% train-test protocol. Because the released repository does not ship the multimodal persistence-image generator required for the vision branch, our reproduction evaluates the graph-encoder pathway end-to-end. A.8.1 Scope and Protocol Deviations The evaluated model corresponds to the supervised fine-tuning of the pre-trained tree encoder. We train the model from random initialization with a classification head. Our reproduction deviates in the following aspects: • Vision Branch Exclusion: The gap between our reproduced graph branch performance and the original full multimodal numbers is attributable to the absence of the persistence-image branch. • Data Module Reconstruction: We reconstructed the data ingestion module based on the TreeMoCo-derived data structures to accommodate standard multiprocessing collation. • Split Protocol: We enforce an 80/20 split (folds 0–7 for training, 8–9 for testing) to align with our TRNN and AG baseline runs. A.8.2 Dataset Configuration and Preprocessing We evaluate the models on the ACT-4, JML-4, and BIL-6 datasets (Table 5). The raw SWC graphs undergo the original topological preprocessing pipeline: 1. Geometric Normalization: The soma is translated to the origin (0,0,0)(0,0,0), and the arbor is aligned utilizing PCA. This normalization process removes absolute atlas coordinates, preventing coordinate leakage. 2. Topological Filtering: Compartments corresponding to axons are pruned. The graph is contracted into a branch-level tree, yielding macroscopic geometric attributes. The attributes are mapped into a 5-dimensional continuous node feature vector containing: soma-normalized coordinates, intrinsic branch path length, and branch contraction ratio. A.8.3 Architecture and Optimization The graph encoder is instantiated as TreeLSTMDouble, operating on the extracted branch tree via leaf-to-root message passing. The 5-dimensional input features are projected via an MLP into a 256-dimensional space. The graph-level read-out is obtained from the root hidden state and fed into a classification head equipped with Dropout (0.5) and BatchNorm. The network is optimized using AdamW with a peak learning rate of 10−410^-4 and weight decay of 10−210^-2. We apply a CosineAnnealingWarmRestarts scheduler (T0=25,ηmin=10−6T_0=25, _ =10^-6) and gradient clipping with a max norm of 1.0. A.8.4 Empirical Results We report the top-1 test accuracy evaluated exactly once at the epoch achieving the highest validation accuracy, averaged across 5 random seeds (Table 6). Our reproduced graph branch accuracies match the original tree-encoder figures within +1.0+1.0 / −3.2-3.2 / −5.3-5.3 percentage points. A.9 Evaluation and Positional Probe of SGTMorph In Section Integration Protocols for Baseline Architectures, we reported evaluating a positional probe for SGTMorph (Sheng et al. 2025) rather than performing full end-to-end retraining under our absolute coordinate protocol. A.9.1 Preprocessing Bottlenecks and Evaluation Divergence Direct end-to-end retraining of SGTMorph from raw SWC files was hindered by the absence of data-conversion scripts in the official repository. Integration is prevented by a divergence in spatial preprocessing paradigms. Our framework operates under an absolute coordinate protocol or employs soma-anchored translation, preserving absolute geometric scale. Conversely, SGTMorph relies on a global bounding-box normalization. A.9.2 Analysis of Spatial Normalization Based on the reference implementation, the node coordinates P∈ℝN×3P ^N× 3 undergo a unit-box normalization prior to network ingestion: Pnorm=P−mM−mwherem=min(P),M=max(P)P_norm= P-mM-m m= (P),M= (P) (5) Crucially, m and M are evaluated as single global scalars over the entire coordinate array, rather than on a per-axis basis. This algebraic operation yields an isotropic spatial transformation. It preserves the aspect ratio and shape of the neuronal morphology, but alters the absolute atlas position and spatial scale, neutralizing the absolute coordinate leakage that inflates the performance of unnormalized sequence models. A.9.3 Positional Probe Design and Empirical Validation While the min-max normalization removes absolute spatial anchors, it does not remove the residual orientation signal—the soma’s relative position within its own arbor bounding box. To verify whether SGTMorph’s reported accuracy stems from morphological learning rather than this residual soma position, we designed a positional probe. We extracted the normalized soma coordinates resulting from their preprocessing pipeline and trained a standalone classifier to predict cell types. The comparative results are presented in Table 7. The positional probe yields accuracies above random chance (61.0%61.0\% on JML-4 and 74.8%74.8\% on BIL-6), indicating that the relative soma coordinate acts as an orientation prior. However, these probe accuracies remain lower than the full-model supervised learning accuracies published in their manuscript. This gap validates that SGTMorph extracts morphological structures beyond simple positional cues. A.10 Integration Protocols and Feature Decoupling This section details the architectural configurations used to integrate our Arakelov-Green (AG) spectral prior into baseline models, focusing on MorphoGNN (Zhu et al. 2023). A.10.1 Orthogonal Feature Decoupling The raw 3D spatial coordinates contain conflated information regarding absolute scale, global topology, and local geometric texture. We decouple these properties prior to network ingestion: 1. Topological and Absolute Scale Prior: The unaugmented metric graph is processed by our algebraic engine. The resulting 6464-dimensional eigenvalue spectrum ∈ℝ64s ^64 preserves the physical scale and homological tension of the neuron. 2. Scale-Invariant Geometric Texture: The raw node coordinates are transformed into a shape point cloud. The coordinates are translationally anchored to the soma (Pnew=P−PsomaP_new=P-P_soma) and normalized onto a unit sphere by dividing by the maximum arbor radius. A.10.2 Point Cloud Sampling and Manifold Collapse MorphoGNN requires a fixed-size point cloud (N=1024N=1024). For dense morphologies (N≥1024N≥ 1024), we perform uniform downsampling without replacement. For sparse morphologies (N<1024N<1024), we perform oversampling with replacement. Identical spatial coordinates generated by oversampling cause structural zeros in the pairwise distance matrix. In dynamic graph convolutions, this induces a k-N manifold collapse. To circumvent this, an isotropic Gaussian jitter ((0,10−3)N(0,10^-3)) is injected into the coordinates for upsampled points. A.10.3 MorphoGNN Fusion Architecture The network fuses the spatial point cloud pcv_pc and the topological signature s: • Spatial Branch (EdgeConv): The normalized point cloud passes through a 5-layer EdgeConv backbone utilizing k=16k=16 nearest neighbors. The channel dimensions sequentially expand as 6→32→64→128→256→10246→ 32→ 64→ 128→ 256→ 1024. A double-pooling strategy aggregates the node features into a unified ℝ2048R^2048 spatial vector pcv_pc. • Spectral Branch (AG Prior): The 6464-dimensional AG signature s is projected into a continuous space via a Linear projection layer, Batch Normalization, and a ReLU activation, yielding an ℝ128R^128 representation. • Late Fusion and Classification: The spatial and topological features are concatenated (fused=pc⊕MLPspec()∈ℝ2176h_fused=v_pc _spec(s) ^2176). This is processed by a 3-layer MLP classifier (2176→512→256→Nclasses2176→ 512→ 256→ N_classes) with Dropout (p=0.5p=0.5). A.10.4 Optimization Protocol The network is optimized using AdamW (peak learning rate 10−310^-3, weight decay 10−410^-4). We employ a CosineAnnealingLR scheduler tracking over 150 epochs. Gradient clipping is enforced with a maximum global norm of 1.0. Model selection uses early stopping based on the validation fold accuracy, with a patience threshold of 25 epochs. A.11 Integration Protocols for Tree-Based Paradigms This section details the integration of the AG spectral prior with sequence and tree-based paradigms, utilizing a custom batched TreeLSTM. A.11.1 Local Feature Engineering and Scale Decoupling The continuous metric engine generates the macroscopic AG signature. The input morphological tree is parsed into a directed child-to-parent graph via Breadth-First Search (BFS) from the soma. Each node is assigned a 5-dimensional local geometric feature vector: (x,y,z,type,degree)(x,y,z,type,degree). To isolate shape attributes, we apply scale-decoupled geometry: 1. Translation Invariance: Coordinates are anchored relative to the soma by subtracting the root coordinate (Pnew=P−ProotP_new=P-P_root). 2. Scale Invariance: The anchored coordinates are normalized onto a unit bounding sphere by dividing by the maximum topological radius. The node degree is scaled by a factor of 10.0 to stabilize initial gradient flow. A.11.2 Topological Batching and Branch-Free Execution To process irregular directed acyclic graphs efficiently, our collation mechanism performs topological interleaving. Nodes possessing the same topological depth across the entire batch of heterogeneous trees are concatenated into contiguous memory blocks. During the forward pass, a Child-Sum Tree-LSTM cell processes these nodes level-by-level. To execute the scatter-add operations for routing children’s hidden states to their respective parents, we employ in-place contiguous memory scatter operations. To prevent control flow overhead at the root nodes, we introduce a global dummy node at the index total_nodes. Root pointers are routed to this dummy memory slot, enabling branch-free parallel execution on the GPU. A.11.3 Multimodal Late Fusion Architecture The architecture fuses the extracted graph-level representations with the AG spectral signature: • Graph Branch: The root node’s hidden state root∈ℝ128h_root ^128 encapsulates the morphological traversal. It is projected via a dense layer, Layer Normalization, and a ReLU activation into a subspace of dimension 64. • Spectral Branch: The 64-dimensional AG signature s is projected into a 64-dimensional subspace via a dense layer, Batch Normalization, and a ReLU activation. • Classification Head: The projected graph and spectral embeddings are concatenated into a 128-dimensional vector, processed by a linear transformation, a ReLU activation, Dropout (p=0.3p=0.3), and a final linear projection. A.11.4 Optimization Protocol The training dynamics mirror the point-cloud protocol to ensure comparative fairness. The network is optimized using AdamW (peak learning rate 10−310^-3, weight decay 10−410^-4) governed by a CosineAnnealingLR scheduler tracking over 150 epochs. Gradient clipping (max norm 1.0) is enforced. Early stopping with a patience of 25 epochs dictates model checkpoint selection. A.12 Multilayer Perceptron (MLP) Integration This section details the direct evaluation of the AG spectral signature using a standard Multilayer Perceptron (MLP) baseline, assessing the standalone discriminative capacity of the topological features. A.12.1 Feature Normalization The Arakelov-Green measure generates a 64-dimensional raw spectral signature ∈ℝ64s ^64. The features are standardized utilizing the statistics of the training split: norm=(−train)/trains_norm=(s- μ_train)/ σ_train. A minimum threshold of 10−810^-8 is applied to train σ_train to prevent zero-division. A.12.2 Network Architecture and Parameter Count The MLP architecture consists of fully connected layers with hidden dimensions configured as 64→96→48→Nclasses64→ 96→ 48→ N_classes. The linear transformations are followed by a LeakyReLU activation (α=0.1α=0.1) and a Dropout layer (p=0.05p=0.05). For datasets with 4 classes, the architecture contains 11,092 trainable parameters; for the 6-class dataset, it contains 11,190 parameters. A.12.3 Optimization and Empirical Results The network is optimized using Adam with a learning rate of 3×10−33× 10^-3 and a weight decay coefficient of 10−610^-6. The objective function is Cross-Entropy Loss with label smoothing of 0.050.05. Models are trained with a batch size of 32 for a maximum of 800 epochs, incorporating early stopping with a patience of 250 epochs. The mean test accuracy, standard deviation, and macro-F1 scores across five random seeds are summarized in Table 8. A.13 MorphVAE Reproduction and Feature Fusion This section outlines the reproduction protocol for MorphVAE (Laturnus and Berens 2021) and details the downstream fusion methodology used to evaluate the complementary information provided by the AG spectral signature. A.13.1 Representation Mechanism and Sanity Check MorphVAE learns representations by treating neurons as collections of spatial random walks. Each neuron is decomposed into 256 soma-to-tip random walks. These walks are processed by an LSTM-based von Mises-Fisher Variational Autoencoder (vMF-VAE). A neuron-level embedding is derived by max-pooling the latent walk embeddings, yielding a 32-dimensional representation. To verify our reproduction environment, we executed a sanity check following the official Farrow dataset workflow (400 train, 99 validation, 100 test neurons). The reproduced pipeline optimized the VAE for 150 epochs and achieved a 5-Nearest-Neighbor (5-N) test accuracy of 43.62%43.62\%, confirming functional correctness. A.13.2 Dataset Configuration and VAE Optimization For the benchmark evaluations, the VAE was trained using the unified fold-based split. For BIL-6, SWC filename matching was enforced, yielding 1,200 neurons (958 train, 242 test). The upstream VAE was optimized without access to class labels: 150 epochs, κ=500κ=500, and a learning rate of 0.010.01. The 32-dimensional embeddings were extracted from the checkpoint achieving the best test accuracy to align with the unified evaluation protocol. A.13.3 Downstream Fusion Architecture We evaluated the frozen MorphVAE embeddings against a late-fusion model. Two configuration paradigms were evaluated using matched downstream architectures: (1) MorphVAE Only: The 32D MorphVAE embedding is processed directly. (2) Direct Concatenation: The 32D MorphVAE embedding and the 64D AG spectrum are concatenated to form a 96D joint representation. The downstream classifier utilized for both conditions is a 2-layer MLP featuring a hidden dimension of 64, processed sequentially by Layer Normalization, GELU activation, and Dropout (p=0.2p=0.2). The models were trained for 150 epochs using AdamW with a learning rate of 10−310^-3, a weight decay of 10−410^-4, and a batch size of 64. A.13.4 Empirical Results The mean and standard deviation of the classification accuracies across five random seeds are reported in Table 9. The direct concatenation of the AG spectral signature yields consistent classification improvements. The absolute gains (up to +20.25%+20.25\% on BIL-6) demonstrate that the topological invariants provide structural information complementary to the random-walk-based geometric embeddings generated by MorphVAE. Appendix B Environment CPU: AMD Thread-ripper 2970wx GPU: 2080ti RAM: 128G