{"paper":{"title":"On the Origin of Toric Diagrams","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"hep-th","authors_text":"Diego Rodr\\'iguez-G\\'omez, Sebasti\\'an Franco","submitted_at":"2026-07-21T18:00:00Z","abstract_excerpt":"Five-dimensional superconformal field theories ($5d$ SCFTs) can be encoded by Generalized Toric Polygons (GTPs), where external legs of the dual $(p,q)$ five-brane web correspond to $T$-cones. Hanany-Witten transitions act on these geometries by flipping $T$-cones about their apex, thereby naturally endowing the choice of origin in the polygon with physical significance. It was recently conjectured that a suitably graded Hilbert series equals the Ehrhart series of the dual polytope, which, in turn, is an invariant under such mutations. In this paper, we introduce a prescription for assigning s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.19460","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.19460/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}