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We introduce HSG-12M, the first large-scale dataset of $\\textbf{spatial multigraphs}-$graphs embedded in a metric space where multiple geometrically distinct trajectories between two nodes are retained as separate edges. HSG-12M contains 11.6 million static and 5.1 million dynamic $\\textit{Hamiltonian spectral graphs}$ across 1401 characteristic-polynomial classes, derived from 177 TB of spectral potential data. Each graph encodes the full geometry of a 1-D crystal's energy spectrum on the complex plane, producing diverse, physics-grounded topologies that transcend conventional node-coordinate datasets. To enable future extensions, we release $\\texttt{Poly2Graph}$: a high-performance, open-source pipeline that maps arbitrary 1-D crystal Hamiltonians to spectral graphs. Benchmarks with popular GNNs expose new challenges in learning from multi-edge geometry at scale. Beyond its practical utility, we show that spectral graphs serve as universal topological fingerprints of polynomials, vectors, and matrices, forging a new algebra-to-graph link. HSG-12M lays the groundwork for geometry-aware graph learning and new opportunities of data-driven scientific discovery in condensed matter physics and beyond.","url_abs":"https://arxiv.org/abs/2506.08618v1","url_pdf":"https://arxiv.org/pdf/2506.08618v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"2506-08618","repo_url":"https://github.com/sarinstein-yan/hsg-12m","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"pytorch","reach":{"status":"ok","spdx":"Apache-2.0"}},{"paper_slug":"2506-08618","repo_url":"https://github.com/sarinstein-yan/poly2graph","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"pytorch","reach":null}],"tasks":[{"task_slug":"graph-learning","task_name":"Graph Learning"},{"task_slug":"scientific-discovery","task_name":"scientific discovery"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2506.08618","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2506.08618"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. 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