{"paper":{"title":"Discrete Einstein metrics on trees","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"Discrete Einstein metrics exist and are unique on trees under Lin-Lu-Yau curvature, but positive-curvature cases require caterpillar trees.","cross_cats":[],"primary_cat":"math.DG","authors_text":"Bobo Hua, Haoxuan Cheng, Shuliang Bai","submitted_at":"2026-04-24T11:07:15Z","abstract_excerpt":"We establish the existence and uniqueness of discrete Einstein metrics on trees under Lin-Lu-Yau Ricci curvature using Perron-Frobenius theory. We establish a sharp upper bound for the largest eigenvalue of the associated Ricci matrix in terms of the maximum degree. Turning to structural properties, notably, the existence of a positive-curvature Einstein metric implies the tree must be a caterpillar. Furthermore, these metrics exhibit radial monotonicity, with edge weights decreasing strictly away from the maximal edge."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We establish the existence and uniqueness of discrete Einstein metrics on trees under Lin-Lu-Yau Ricci curvature using Perron-Frobenius theory. Notably, the existence of a positive-curvature Einstein metric implies the tree must be a caterpillar. Furthermore, these metrics exhibit radial monotonicity, with edge weights decreasing strictly away from the maximal edge.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The curvature operator constructed from the Lin-Lu-Yau Ricci curvature on the tree must satisfy the positivity or irreducibility conditions required for Perron-Frobenius theory to guarantee a unique positive eigenvector; the abstract does not specify how this is verified or what happens if the tree has vertices of high degree that break positivity.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Existence and uniqueness of discrete Einstein metrics on trees under Lin-Lu-Yau Ricci curvature is established via Perron-Frobenius theory, with positive curvature possible only on caterpillar trees and edge weights decreasing radially from the maximal edge.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Discrete Einstein metrics exist and are unique on trees under Lin-Lu-Yau curvature, but positive-curvature cases require caterpillar trees.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"fd1026ced139d4dbccbbf7dcb5e3d8f94a9ae67c463d6b00c1080e65cc4d7866"},"source":{"id":"2604.22449","kind":"arxiv","version":2},"verdict":{"id":"8ef076e2-da0b-48ea-bdf8-82db501e93eb","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T09:49:16.611794Z","strongest_claim":"We establish the existence and uniqueness of discrete Einstein metrics on trees under Lin-Lu-Yau Ricci curvature using Perron-Frobenius theory. Notably, the existence of a positive-curvature Einstein metric implies the tree must be a caterpillar. Furthermore, these metrics exhibit radial monotonicity, with edge weights decreasing strictly away from the maximal edge.","one_line_summary":"Existence and uniqueness of discrete Einstein metrics on trees under Lin-Lu-Yau Ricci curvature is established via Perron-Frobenius theory, with positive curvature possible only on caterpillar trees and edge weights decreasing radially from the maximal edge.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The curvature operator constructed from the Lin-Lu-Yau Ricci curvature on the tree must satisfy the positivity or irreducibility conditions required for Perron-Frobenius theory to guarantee a unique positive eigenvector; the abstract does not specify how this is verified or what happens if the tree has vertices of high degree that break positivity.","pith_extraction_headline":"Discrete Einstein metrics exist and are unique on trees under Lin-Lu-Yau curvature, but positive-curvature cases require caterpillar trees."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.22449/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T10:40:13.628665Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T23:57:27.590351Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"7fd1e83628616eadc9a13aedf29fcb28f6088bce6347b60562b2c73de4d09e82"},"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"}