{"paper":{"title":"Scaffolds for Higher Tropical Grassmannians: Foundations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Any integer positive tropical Plucker vector has a unique representation as a normal CAT(0) planar graph.","cross_cats":["hep-th","math.AG"],"primary_cat":"math.CO","authors_text":"Nick Early, Thomas Lam","submitted_at":"2026-04-28T04:30:41Z","abstract_excerpt":"Scaffolds are the one-dimensional skeleta of high-dimensional flag simplicial complexes of nonpositive curvature. They generalize the phylogenetic trees of Trop G(2,n) to arbitrary $k$, drawing together SL(k)-web bases, affine buildings, the combinatorics of the positive tropical Grassmannian and low-dimensional topology. We prove that scaffolds model points in all tropical Grassmannians via a $k$-point distance function.\n  In this paper, we study in detail CAT(0) planar graphs, which are positive scaffolds for the tropical Grassmannian of three-planes. CAT(0) planar graphs are directed versio"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that scaffolds model points in all tropical Grassmannians via a k-point distance function. Our main result is the construction of a unique representation of any given integer positive tropical Plucker vector by a normal CAT(0) planar graph.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the defined normal CAT(0) planar graphs and their strand combinatorics faithfully capture the tropical Plucker vectors and basis expansions without hidden conditions or post-hoc adjustments to the graph properties.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Scaffolds and normal CAT(0) planar graphs provide unique combinatorial representations of positive tropical Plucker vectors and embed into tropical linear spaces and affine buildings.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Any integer positive tropical Plucker vector has a unique representation as a normal CAT(0) planar graph.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"c78e7d5fa4fae36db0aae9111cb32d03f28765f3a4b6ec11bd8d3ba35fb0b112"},"source":{"id":"2604.25211","kind":"arxiv","version":1},"verdict":{"id":"6599a7e6-e903-4d8f-8790-4a2c9d707d52","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-07T15:55:45.270512Z","strongest_claim":"We prove that scaffolds model points in all tropical Grassmannians via a k-point distance function. Our main result is the construction of a unique representation of any given integer positive tropical Plucker vector by a normal CAT(0) planar graph.","one_line_summary":"Scaffolds and normal CAT(0) planar graphs provide unique combinatorial representations of positive tropical Plucker vectors and embed into tropical linear spaces and affine buildings.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the defined normal CAT(0) planar graphs and their strand combinatorics faithfully capture the tropical Plucker vectors and basis expansions without hidden conditions or post-hoc adjustments to the graph properties.","pith_extraction_headline":"Any integer positive tropical Plucker vector has a unique representation as a normal CAT(0) planar graph."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.25211/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T05:35:01.693160Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T21:23:08.180554Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"337486eb511005a03b5d753aa8cc8a84b0d4d6e009f1c8aeee71c54906a62fd1"},"references":{"count":43,"sample":[{"doi":"","year":2020,"title":"Akhmejanov, Non-elliptic webs and convex sets in the affine building","work_id":"2bbb8837-7579-457f-9e34-028cf5698635","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2016,"title":"F. Ardila, F. Rinc\\'on, and L. Williams, Positroids and non-crossing partitions. Trans. Amer. Math. Soc. 368 (2016), 337--363","work_id":"445017a1-a814-4357-acbc-844c61cb2597","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"N. Arkani-Hamed, T. Lam, and M. Spradlin, Positive configuration space. Comm.\\ Math.\\ Phys. \\ 384 (2021), 909--954","work_id":"c5d25d42-967d-486c-bd47-9806dc673667","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"N. Arkani-Hamed, T. Lam, and M. Spradlin, Non-perturbative geometries for planar N = 4 SYM amplitudes. JHEP (2021), article number 065","work_id":"20769070-4d33-4383-972a-21b24044d1dc","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2019,"title":"F. Cachazo, N. Early, A. Guevara, and S. Mizera, Scattering equations: from projective spaces to tropical Grassmannians. 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