{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:5XCFCG2F64CMKCI6TANERA5BR4","short_pith_number":"pith:5XCFCG2F","canonical_record":{"source":{"id":"1012.3905","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-12-16T20:41:37Z","cross_cats_sorted":[],"title_canon_sha256":"b2715146a2ffb7ca1da532c0cac95eb91754717f6bc038ebc8c94f9e3aee1dbc","abstract_canon_sha256":"835b0d3f9784acbc39255a0217313e001af13a9c05ba25141647a603c9a7e3a8"},"schema_version":"1.0"},"canonical_sha256":"edc4511b45f704c5091e981a4883a18f1e48d99fd74d6c674cd89042ac5a0abf","source":{"kind":"arxiv","id":"1012.3905","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.3905","created_at":"2026-05-18T02:32:31Z"},{"alias_kind":"arxiv_version","alias_value":"1012.3905v3","created_at":"2026-05-18T02:32:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.3905","created_at":"2026-05-18T02:32:31Z"},{"alias_kind":"pith_short_12","alias_value":"5XCFCG2F64CM","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"5XCFCG2F64CMKCI6","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"5XCFCG2F","created_at":"2026-05-18T12:26:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:5XCFCG2F64CMKCI6TANERA5BR4","target":"record","payload":{"canonical_record":{"source":{"id":"1012.3905","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-12-16T20:41:37Z","cross_cats_sorted":[],"title_canon_sha256":"b2715146a2ffb7ca1da532c0cac95eb91754717f6bc038ebc8c94f9e3aee1dbc","abstract_canon_sha256":"835b0d3f9784acbc39255a0217313e001af13a9c05ba25141647a603c9a7e3a8"},"schema_version":"1.0"},"canonical_sha256":"edc4511b45f704c5091e981a4883a18f1e48d99fd74d6c674cd89042ac5a0abf","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:32:31.296655Z","signature_b64":"QJnsytT1oDIEf39Y5WzEu6IgDcXGA0IpshuJT4Z/bkqHnjuJQT+RDJ3iByecOGXyGfXPI9IB06g2/4CFKPy3Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"edc4511b45f704c5091e981a4883a18f1e48d99fd74d6c674cd89042ac5a0abf","last_reissued_at":"2026-05-18T02:32:31.296287Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:32:31.296287Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1012.3905","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":"2026-05-18T02:32:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8CTbfo5YVYzh2rhMK8SsD+LQDHbik5CvwfYN5YGo4XK8yjpWOeDuaKhD/dhBtbATSThvx78SKMahIu4lNN8/Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T12:25:03.155704Z"},"content_sha256":"13581598af32b65de0bdf57bb0c8e08726103900b98106ca9ff79dd6de301dac","schema_version":"1.0","event_id":"sha256:13581598af32b65de0bdf57bb0c8e08726103900b98106ca9ff79dd6de301dac"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:5XCFCG2F64CMKCI6TANERA5BR4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Realizability of Polytopes as a Low Rank Matrix Completion Problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Michael Gene Dobbins","submitted_at":"2010-12-16T20:41:37Z","abstract_excerpt":"This article gives necessary and sufficient conditions for a relation to be the containment relation between the facets and vertices of a polytope. Also given here, are a set of matrices parameterizing the linear moduli space and another set parameterizing the projective moduli space of a combinatorial polytope."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.3905","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":"2026-05-18T02:32:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8W9+P7zq1GhUfcUpb8PUZ3cRsHTMopeYZJpH8AAXhddmdk9yJAmOi6EET4Et5LnbOIpVBQ0HF3cWUNNLMcB5BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T12:25:03.157610Z"},"content_sha256":"42d8e0dcf880d23fcdcb9dc09a25ff7172964d4eac7564cf915569a6084d8875","schema_version":"1.0","event_id":"sha256:42d8e0dcf880d23fcdcb9dc09a25ff7172964d4eac7564cf915569a6084d8875"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5XCFCG2F64CMKCI6TANERA5BR4/bundle.json","state_url":"https://pith.science/pith/5XCFCG2F64CMKCI6TANERA5BR4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5XCFCG2F64CMKCI6TANERA5BR4/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":"2026-05-23T12:25:03Z","links":{"resolver":"https://pith.science/pith/5XCFCG2F64CMKCI6TANERA5BR4","bundle":"https://pith.science/pith/5XCFCG2F64CMKCI6TANERA5BR4/bundle.json","state":"https://pith.science/pith/5XCFCG2F64CMKCI6TANERA5BR4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5XCFCG2F64CMKCI6TANERA5BR4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:5XCFCG2F64CMKCI6TANERA5BR4","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":"835b0d3f9784acbc39255a0217313e001af13a9c05ba25141647a603c9a7e3a8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-12-16T20:41:37Z","title_canon_sha256":"b2715146a2ffb7ca1da532c0cac95eb91754717f6bc038ebc8c94f9e3aee1dbc"},"schema_version":"1.0","source":{"id":"1012.3905","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.3905","created_at":"2026-05-18T02:32:31Z"},{"alias_kind":"arxiv_version","alias_value":"1012.3905v3","created_at":"2026-05-18T02:32:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.3905","created_at":"2026-05-18T02:32:31Z"},{"alias_kind":"pith_short_12","alias_value":"5XCFCG2F64CM","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"5XCFCG2F64CMKCI6","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"5XCFCG2F","created_at":"2026-05-18T12:26:04Z"}],"graph_snapshots":[{"event_id":"sha256:42d8e0dcf880d23fcdcb9dc09a25ff7172964d4eac7564cf915569a6084d8875","target":"graph","created_at":"2026-05-18T02:32:31Z","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":"This article gives necessary and sufficient conditions for a relation to be the containment relation between the facets and vertices of a polytope. Also given here, are a set of matrices parameterizing the linear moduli space and another set parameterizing the projective moduli space of a combinatorial polytope.","authors_text":"Michael Gene Dobbins","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-12-16T20:41:37Z","title":"Realizability of Polytopes as a Low Rank Matrix Completion Problem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.3905","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":"sha256:13581598af32b65de0bdf57bb0c8e08726103900b98106ca9ff79dd6de301dac","target":"record","created_at":"2026-05-18T02:32:31Z","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":"835b0d3f9784acbc39255a0217313e001af13a9c05ba25141647a603c9a7e3a8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-12-16T20:41:37Z","title_canon_sha256":"b2715146a2ffb7ca1da532c0cac95eb91754717f6bc038ebc8c94f9e3aee1dbc"},"schema_version":"1.0","source":{"id":"1012.3905","kind":"arxiv","version":3}},"canonical_sha256":"edc4511b45f704c5091e981a4883a18f1e48d99fd74d6c674cd89042ac5a0abf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"edc4511b45f704c5091e981a4883a18f1e48d99fd74d6c674cd89042ac5a0abf","first_computed_at":"2026-05-18T02:32:31.296287Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:32:31.296287Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QJnsytT1oDIEf39Y5WzEu6IgDcXGA0IpshuJT4Z/bkqHnjuJQT+RDJ3iByecOGXyGfXPI9IB06g2/4CFKPy3Cw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:32:31.296655Z","signed_message":"canonical_sha256_bytes"},"source_id":"1012.3905","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:13581598af32b65de0bdf57bb0c8e08726103900b98106ca9ff79dd6de301dac","sha256:42d8e0dcf880d23fcdcb9dc09a25ff7172964d4eac7564cf915569a6084d8875"],"state_sha256":"f496f94e8fd9fb2386be39114ec51fd861ce88407c850b46201c392e20882219"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/BPxnSNRyNd3vxdamTZV4VoDLsvGW/WLGDCbUTyYX2GUELr2SGEMVkzEnMbwuzQP9YmGLUVKW1gaRi/1lu/iAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-23T12:25:03.162023Z","bundle_sha256":"0fcb3382dca70eb1e6385fc173f2f327c84caf4c3522372a6dba6374b2a39a2e"}}