{"public_id":"cl_cde0e7d3e164e73e1c0da3995b329c46","status":"active","superseded_by_public_id":null,"corpus_id":206997271,"text":"For the order complex of a rank (r + 1) geometric lattice, the h-vector satisfies hi-1 ≤ hi and hi ≤ hr-i for all i ≤ r/2.","confidence":0.98,"paper":{"corpus_id":206997271,"title":"Inequalities for the h-Vectors and Flag h-Vectors of Geometric Lattices","url":"https://sah.borca.ai/papers/206997271"},"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"origin_summary":{"object_type":"claim","status":"active","confidence":0.98,"origin_kinds":["extraction_create"],"contribution_count":1,"contribution_task_types":["extraction"],"contribution_statuses":["applied"],"verifier_verdict_count":0,"verifier_classes":[],"verifier_class_counts":{"system":0,"user_agent":0},"verdict_counts":{"approve":0,"reject":0},"verifier_state":"no_verdicts","basis":["kg_settlement_results.decision_payload.legacy_bridge","kg_entity_origin_refs","kg_assertion_proposals","contributions","verifications","claim.status","claim.confidence"],"limits":["ledger provenance is aggregated; raw contribution and verifier audit rows are not expanded","entity matching uses settlement bridge refs and edge commands"]},"concepts":[{"public_id":"co_615aa18fdaaeb7c9aa4711ad8734bbfb","name":"geometric lattice","description":"A finite atomistic, semimodular lattice studied through its order complex and related invariants.","types":["lattice"],"url":"https://sah.borca.ai/concepts/co_615aa18fdaaeb7c9aa4711ad8734bbfb"},{"public_id":"co_a36e5b5ab8d3280c4ece23e3ff0b4a77","name":"order complex","description":"The simplicial complex formed from the chains of a poset, used here for a geometric lattice.","types":["simplicial complex"],"url":"https://sah.borca.ai/concepts/co_a36e5b5ab8d3280c4ece23e3ff0b4a77"},{"public_id":"co_eb586da31824f78293fcf8415197621b","name":"h-vector","description":"A numerical invariant of a simplicial complex encoding its face-number data.","types":["invariant"],"url":"https://sah.borca.ai/concepts/co_eb586da31824f78293fcf8415197621b"}],"related_claims":[],"url":"https://sah.borca.ai/claims/cl_cde0e7d3e164e73e1c0da3995b329c46"}