{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:YMVBYGV2JBOR3WZZ2APFFVLHKD","short_pith_number":"pith:YMVBYGV2","schema_version":"1.0","canonical_sha256":"c32a1c1aba485d1ddb39d01e52d56750f69c01b94aea6ab7614cffa6e15f53e6","source":{"kind":"arxiv","id":"2604.22449","version":2},"attestation_state":"computed","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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2604.22449","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.DG","submitted_at":"2026-04-24T11:07:15Z","cross_cats_sorted":[],"title_canon_sha256":"1ae7042af443ce9f98382cba2983eec062717b80b5f78e96471b2766d3d359b7","abstract_canon_sha256":"600f05a3b92f95ecede91eee7858f723fe844393c90cc3625f2ad2efa9cca63c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-25T02:01:20.924630Z","signature_b64":"ZgC/ggFWXYu4vuMOehwV25zlsKrsYZckNUasUubHys3DC3skOT/0L+VAOErRdkbMofTzQo3JIffzAIEXqca2DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c32a1c1aba485d1ddb39d01e52d56750f69c01b94aea6ab7614cffa6e15f53e6","last_reissued_at":"2026-05-25T02:01:20.923816Z","signature_status":"signed_v1","first_computed_at":"2026-05-25T02:01:20.923816Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2604.22449","created_at":"2026-05-25T02:01:20.923935+00:00"},{"alias_kind":"arxiv_version","alias_value":"2604.22449v2","created_at":"2026-05-25T02:01:20.923935+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.22449","created_at":"2026-05-25T02:01:20.923935+00:00"},{"alias_kind":"pith_short_12","alias_value":"YMVBYGV2JBOR","created_at":"2026-05-25T02:01:20.923935+00:00"},{"alias_kind":"pith_short_16","alias_value":"YMVBYGV2JBOR3WZZ","created_at":"2026-05-25T02:01:20.923935+00:00"},{"alias_kind":"pith_short_8","alias_value":"YMVBYGV2","created_at":"2026-05-25T02:01:20.923935+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2605.20862","citing_title":"A Classification of Positive-Curvature Discrete Einstein Metrics on Trees","ref_index":24,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/YMVBYGV2JBOR3WZZ2APFFVLHKD","json":"https://pith.science/pith/YMVBYGV2JBOR3WZZ2APFFVLHKD.json","graph_json":"https://pith.science/api/pith-number/YMVBYGV2JBOR3WZZ2APFFVLHKD/graph.json","events_json":"https://pith.science/api/pith-number/YMVBYGV2JBOR3WZZ2APFFVLHKD/events.json","paper":"https://pith.science/paper/YMVBYGV2"},"agent_actions":{"view_html":"https://pith.science/pith/YMVBYGV2JBOR3WZZ2APFFVLHKD","download_json":"https://pith.science/pith/YMVBYGV2JBOR3WZZ2APFFVLHKD.json","view_paper":"https://pith.science/paper/YMVBYGV2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2604.22449&json=true","fetch_graph":"https://pith.science/api/pith-number/YMVBYGV2JBOR3WZZ2APFFVLHKD/graph.json","fetch_events":"https://pith.science/api/pith-number/YMVBYGV2JBOR3WZZ2APFFVLHKD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YMVBYGV2JBOR3WZZ2APFFVLHKD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YMVBYGV2JBOR3WZZ2APFFVLHKD/action/storage_attestation","attest_author":"https://pith.science/pith/YMVBYGV2JBOR3WZZ2APFFVLHKD/action/author_attestation","sign_citation":"https://pith.science/pith/YMVBYGV2JBOR3WZZ2APFFVLHKD/action/citation_signature","submit_replication":"https://pith.science/pith/YMVBYGV2JBOR3WZZ2APFFVLHKD/action/replication_record"}},"created_at":"2026-05-25T02:01:20.923935+00:00","updated_at":"2026-05-25T02:01:20.923935+00:00"}