{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:OTXYAVUBPWU77H42Y547L2VCGZ","short_pith_number":"pith:OTXYAVUB","canonical_record":{"source":{"id":"0909.0694","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-09-03T15:58:13Z","cross_cats_sorted":[],"title_canon_sha256":"c742e3177fb6d06f39b30544108e4a26dc961ee6f4405f576e1bd2e2f4acf5ad","abstract_canon_sha256":"6215dac3305472f832beeb0809fb61d64eb1256a52eaf4fcb0400c21f4dfac9f"},"schema_version":"1.0"},"canonical_sha256":"74ef8056817da9ff9f9ac779f5eaa23663bf66cb1dddf96fee47d5eaea131c9e","source":{"kind":"arxiv","id":"0909.0694","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0909.0694","created_at":"2026-05-18T04:16:10Z"},{"alias_kind":"arxiv_version","alias_value":"0909.0694v3","created_at":"2026-05-18T04:16:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0909.0694","created_at":"2026-05-18T04:16:10Z"},{"alias_kind":"pith_short_12","alias_value":"OTXYAVUBPWU7","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"OTXYAVUBPWU77H42","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"OTXYAVUB","created_at":"2026-05-18T12:26:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:OTXYAVUBPWU77H42Y547L2VCGZ","target":"record","payload":{"canonical_record":{"source":{"id":"0909.0694","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-09-03T15:58:13Z","cross_cats_sorted":[],"title_canon_sha256":"c742e3177fb6d06f39b30544108e4a26dc961ee6f4405f576e1bd2e2f4acf5ad","abstract_canon_sha256":"6215dac3305472f832beeb0809fb61d64eb1256a52eaf4fcb0400c21f4dfac9f"},"schema_version":"1.0"},"canonical_sha256":"74ef8056817da9ff9f9ac779f5eaa23663bf66cb1dddf96fee47d5eaea131c9e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:16:10.794084Z","signature_b64":"SHPNUEAV0sRJ/ruFkklC9S34bEUv5vw+QqAB9Lb6pgYONW/qfwzAn9OboN+68DS8We+U9E0h7dFEx6cnOrYkBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"74ef8056817da9ff9f9ac779f5eaa23663bf66cb1dddf96fee47d5eaea131c9e","last_reissued_at":"2026-05-18T04:16:10.793456Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:16:10.793456Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0909.0694","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:16:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KQQIitWdtWTXabOOYsqbxo/KLkP0Pk4T4uB/7fpWNWBP2eUl2xaM0Gpp/DBKyRAyhoZWN9Q/Qw6hHMDA7ti0Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T14:17:22.185364Z"},"content_sha256":"2f043ef32279efc4be5175f8ba5778e95ef88d51b060884617e00f9778c734d8","schema_version":"1.0","event_id":"sha256:2f043ef32279efc4be5175f8ba5778e95ef88d51b060884617e00f9778c734d8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:OTXYAVUBPWU77H42Y547L2VCGZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On $\\gamma$-vectors satisfying the Kruskal-Katona inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Eran Nevo, T. Kyle Petersen","submitted_at":"2009-09-03T15:58:13Z","abstract_excerpt":"We present examples of flag homology spheres whose $\\gamma$-vectors satisfy the Kruskal-Katona inequalities. This includes several families of well-studied simplicial complexes, including Coxeter complexes and the simplicial complexes dual to the associahedron and to the cyclohedron. In these cases, we construct explicit simplicial complexes whose $f$-vectors are the $\\gamma$-vectors in question. In another direction, we show that if a flag $(d-1)$-sphere has at most $2d+2$ vertices its $\\gamma$-vector satisfies the Kruskal-Katona inequalities. We conjecture that if $\\Delta$ is a flag homology"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.0694","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:16:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"s4gxJTvhgyOKW51hDzC0RLjmyJXRY1HroCGtPpEX/kVlt9Cb9gLvoJhZKrhw+e6fMHVd+LOusHGTKrzHzvjJBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T14:17:22.185711Z"},"content_sha256":"300995b625ccf0867ed2eff08f8a88b715624109a030c22795a230d53227de60","schema_version":"1.0","event_id":"sha256:300995b625ccf0867ed2eff08f8a88b715624109a030c22795a230d53227de60"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OTXYAVUBPWU77H42Y547L2VCGZ/bundle.json","state_url":"https://pith.science/pith/OTXYAVUBPWU77H42Y547L2VCGZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OTXYAVUBPWU77H42Y547L2VCGZ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-02T14:17:22Z","links":{"resolver":"https://pith.science/pith/OTXYAVUBPWU77H42Y547L2VCGZ","bundle":"https://pith.science/pith/OTXYAVUBPWU77H42Y547L2VCGZ/bundle.json","state":"https://pith.science/pith/OTXYAVUBPWU77H42Y547L2VCGZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OTXYAVUBPWU77H42Y547L2VCGZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:OTXYAVUBPWU77H42Y547L2VCGZ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"6215dac3305472f832beeb0809fb61d64eb1256a52eaf4fcb0400c21f4dfac9f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-09-03T15:58:13Z","title_canon_sha256":"c742e3177fb6d06f39b30544108e4a26dc961ee6f4405f576e1bd2e2f4acf5ad"},"schema_version":"1.0","source":{"id":"0909.0694","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0909.0694","created_at":"2026-05-18T04:16:10Z"},{"alias_kind":"arxiv_version","alias_value":"0909.0694v3","created_at":"2026-05-18T04:16:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0909.0694","created_at":"2026-05-18T04:16:10Z"},{"alias_kind":"pith_short_12","alias_value":"OTXYAVUBPWU7","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"OTXYAVUBPWU77H42","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"OTXYAVUB","created_at":"2026-05-18T12:26:01Z"}],"graph_snapshots":[{"event_id":"sha256:300995b625ccf0867ed2eff08f8a88b715624109a030c22795a230d53227de60","target":"graph","created_at":"2026-05-18T04:16:10Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We present examples of flag homology spheres whose $\\gamma$-vectors satisfy the Kruskal-Katona inequalities. This includes several families of well-studied simplicial complexes, including Coxeter complexes and the simplicial complexes dual to the associahedron and to the cyclohedron. In these cases, we construct explicit simplicial complexes whose $f$-vectors are the $\\gamma$-vectors in question. In another direction, we show that if a flag $(d-1)$-sphere has at most $2d+2$ vertices its $\\gamma$-vector satisfies the Kruskal-Katona inequalities. We conjecture that if $\\Delta$ is a flag homology","authors_text":"Eran Nevo, T. Kyle Petersen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-09-03T15:58:13Z","title":"On $\\gamma$-vectors satisfying the Kruskal-Katona inequalities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.0694","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:2f043ef32279efc4be5175f8ba5778e95ef88d51b060884617e00f9778c734d8","target":"record","created_at":"2026-05-18T04:16:10Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"6215dac3305472f832beeb0809fb61d64eb1256a52eaf4fcb0400c21f4dfac9f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-09-03T15:58:13Z","title_canon_sha256":"c742e3177fb6d06f39b30544108e4a26dc961ee6f4405f576e1bd2e2f4acf5ad"},"schema_version":"1.0","source":{"id":"0909.0694","kind":"arxiv","version":3}},"canonical_sha256":"74ef8056817da9ff9f9ac779f5eaa23663bf66cb1dddf96fee47d5eaea131c9e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"74ef8056817da9ff9f9ac779f5eaa23663bf66cb1dddf96fee47d5eaea131c9e","first_computed_at":"2026-05-18T04:16:10.793456Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:16:10.793456Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SHPNUEAV0sRJ/ruFkklC9S34bEUv5vw+QqAB9Lb6pgYONW/qfwzAn9OboN+68DS8We+U9E0h7dFEx6cnOrYkBA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:16:10.794084Z","signed_message":"canonical_sha256_bytes"},"source_id":"0909.0694","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2f043ef32279efc4be5175f8ba5778e95ef88d51b060884617e00f9778c734d8","sha256:300995b625ccf0867ed2eff08f8a88b715624109a030c22795a230d53227de60"],"state_sha256":"79591382678942848ed676abb554a326202348e861e605d3ece0e19bc1875015"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SklrZJD6V3mWwo/c5/WpjK+cZZRFjC4y2vzIoFebHVnulOPubkbJ0d0odVFRIYuSpXnpb90gK2Ea1Dcv4c7OAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T14:17:22.187670Z","bundle_sha256":"f1d2423fd05a93fb0353fc43bc6088c4589bd52d479c61ac0dfa0d29d4210eb5"}}