{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:P7ZUS4EXXRM3TQVYW6IPKJDKTM","short_pith_number":"pith:P7ZUS4EX","canonical_record":{"source":{"id":"1110.3724","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-17T16:35:11Z","cross_cats_sorted":[],"title_canon_sha256":"51f5eb02801f6a6db64ab5339df1be9cc2d8663812287722144225eb3720aba8","abstract_canon_sha256":"4c64735697e7738e8bc8116ea0900d744114198e802c324156b3e7b141f29b67"},"schema_version":"1.0"},"canonical_sha256":"7ff3497097bc59b9c2b8b790f5246a9b0909ee00da0a9e8cb282a7e702e62a84","source":{"kind":"arxiv","id":"1110.3724","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.3724","created_at":"2026-05-18T04:10:48Z"},{"alias_kind":"arxiv_version","alias_value":"1110.3724v1","created_at":"2026-05-18T04:10:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.3724","created_at":"2026-05-18T04:10:48Z"},{"alias_kind":"pith_short_12","alias_value":"P7ZUS4EXXRM3","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"P7ZUS4EXXRM3TQVY","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"P7ZUS4EX","created_at":"2026-05-18T12:26:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:P7ZUS4EXXRM3TQVYW6IPKJDKTM","target":"record","payload":{"canonical_record":{"source":{"id":"1110.3724","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-17T16:35:11Z","cross_cats_sorted":[],"title_canon_sha256":"51f5eb02801f6a6db64ab5339df1be9cc2d8663812287722144225eb3720aba8","abstract_canon_sha256":"4c64735697e7738e8bc8116ea0900d744114198e802c324156b3e7b141f29b67"},"schema_version":"1.0"},"canonical_sha256":"7ff3497097bc59b9c2b8b790f5246a9b0909ee00da0a9e8cb282a7e702e62a84","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:10:48.622469Z","signature_b64":"cTQmE9Au9x0rBwEWPsHUg3Nr5hd+CZSYkV2SM3kkhWLLU29fHFET4Grg8W724GmPOJ6cCG43OW3RO4rjXUpJDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7ff3497097bc59b9c2b8b790f5246a9b0909ee00da0a9e8cb282a7e702e62a84","last_reissued_at":"2026-05-18T04:10:48.622058Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:10:48.622058Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1110.3724","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:10:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WcpJOCRLqikEW4QT0+BIHTYA4n2QkqW4EKsszP1IsUwDQrgX8AM8dIDjxGtqaNgzHL1oIya0J/tX4jmuQTIcAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T04:10:53.377529Z"},"content_sha256":"ac57c7e6fbb968e8a362f02a24f91ec3106a61abfdbefcc5c90e3fe5b2175449","schema_version":"1.0","event_id":"sha256:ac57c7e6fbb968e8a362f02a24f91ec3106a61abfdbefcc5c90e3fe5b2175449"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:P7ZUS4EXXRM3TQVYW6IPKJDKTM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Unimodality questions for integrally closed lattice polytopes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jan Schepers, Leen Van Langenhoven","submitted_at":"2011-10-17T16:35:11Z","abstract_excerpt":"It is a famous open question whether every integrally closed reflexive polytope has a unimodal Ehrhart delta-vector. We generalize this question to arbitrary integrally closed lattice polytopes and we prove unimodality for the delta-vector of lattice parallelepipeds. This is the first nontrivial class of integrally closed polytopes. Moreover, we suggest a new approach to the problem for reflexive polytopes via triangulations."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.3724","kind":"arxiv","version":1},"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":"2026-05-18T04:10:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/I6rMnXAx0oCIkLEcdPQC7imo1pvcJie/CJK0bGS+OVIKcM9Nfcjn+LVEqk4JRYsNujeBar8+kQI6MIzQjhgBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T04:10:53.377870Z"},"content_sha256":"9fcf101730a4de60e3a69a00651c46c752fcff1e34878aa58df03f632d4406d3","schema_version":"1.0","event_id":"sha256:9fcf101730a4de60e3a69a00651c46c752fcff1e34878aa58df03f632d4406d3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/P7ZUS4EXXRM3TQVYW6IPKJDKTM/bundle.json","state_url":"https://pith.science/pith/P7ZUS4EXXRM3TQVYW6IPKJDKTM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/P7ZUS4EXXRM3TQVYW6IPKJDKTM/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":"2026-06-24T04:10:53Z","links":{"resolver":"https://pith.science/pith/P7ZUS4EXXRM3TQVYW6IPKJDKTM","bundle":"https://pith.science/pith/P7ZUS4EXXRM3TQVYW6IPKJDKTM/bundle.json","state":"https://pith.science/pith/P7ZUS4EXXRM3TQVYW6IPKJDKTM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/P7ZUS4EXXRM3TQVYW6IPKJDKTM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:P7ZUS4EXXRM3TQVYW6IPKJDKTM","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":"4c64735697e7738e8bc8116ea0900d744114198e802c324156b3e7b141f29b67","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-17T16:35:11Z","title_canon_sha256":"51f5eb02801f6a6db64ab5339df1be9cc2d8663812287722144225eb3720aba8"},"schema_version":"1.0","source":{"id":"1110.3724","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.3724","created_at":"2026-05-18T04:10:48Z"},{"alias_kind":"arxiv_version","alias_value":"1110.3724v1","created_at":"2026-05-18T04:10:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.3724","created_at":"2026-05-18T04:10:48Z"},{"alias_kind":"pith_short_12","alias_value":"P7ZUS4EXXRM3","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"P7ZUS4EXXRM3TQVY","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"P7ZUS4EX","created_at":"2026-05-18T12:26:39Z"}],"graph_snapshots":[{"event_id":"sha256:9fcf101730a4de60e3a69a00651c46c752fcff1e34878aa58df03f632d4406d3","target":"graph","created_at":"2026-05-18T04:10:48Z","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":"It is a famous open question whether every integrally closed reflexive polytope has a unimodal Ehrhart delta-vector. We generalize this question to arbitrary integrally closed lattice polytopes and we prove unimodality for the delta-vector of lattice parallelepipeds. This is the first nontrivial class of integrally closed polytopes. Moreover, we suggest a new approach to the problem for reflexive polytopes via triangulations.","authors_text":"Jan Schepers, Leen Van Langenhoven","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-17T16:35:11Z","title":"Unimodality questions for integrally closed lattice polytopes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.3724","kind":"arxiv","version":1},"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:ac57c7e6fbb968e8a362f02a24f91ec3106a61abfdbefcc5c90e3fe5b2175449","target":"record","created_at":"2026-05-18T04:10:48Z","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":"4c64735697e7738e8bc8116ea0900d744114198e802c324156b3e7b141f29b67","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-17T16:35:11Z","title_canon_sha256":"51f5eb02801f6a6db64ab5339df1be9cc2d8663812287722144225eb3720aba8"},"schema_version":"1.0","source":{"id":"1110.3724","kind":"arxiv","version":1}},"canonical_sha256":"7ff3497097bc59b9c2b8b790f5246a9b0909ee00da0a9e8cb282a7e702e62a84","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7ff3497097bc59b9c2b8b790f5246a9b0909ee00da0a9e8cb282a7e702e62a84","first_computed_at":"2026-05-18T04:10:48.622058Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:10:48.622058Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cTQmE9Au9x0rBwEWPsHUg3Nr5hd+CZSYkV2SM3kkhWLLU29fHFET4Grg8W724GmPOJ6cCG43OW3RO4rjXUpJDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:10:48.622469Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.3724","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ac57c7e6fbb968e8a362f02a24f91ec3106a61abfdbefcc5c90e3fe5b2175449","sha256:9fcf101730a4de60e3a69a00651c46c752fcff1e34878aa58df03f632d4406d3"],"state_sha256":"de955fe028d53248f0682cc202438cfd2c1daa7f21a131190e966ef084fca1ea"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eLUNOH3yg+Fprk60iIw8G4+KbjTfMcld+4zYrs23XFuT9m2mREkmU1gjoB6ZMMYoY/GLTPIZN0LiTfffTRo2Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T04:10:53.379763Z","bundle_sha256":"14303961b5fc213d5fbaca5f16ed3cd2258c37d713c43520c644b2a5565f06cc"}}