{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:IEPPPJLNDLQEWCEDWP3XXQWWA6","short_pith_number":"pith:IEPPPJLN","canonical_record":{"source":{"id":"1502.00016","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-30T21:17:27Z","cross_cats_sorted":["quant-ph"],"title_canon_sha256":"831912be0e5029fc768bd6cd7ffafc71702fffe9520bf963c76e4a6ef8b1f4ea","abstract_canon_sha256":"0c36591f8684ef9d9c8c71715ad6cf30333f19be276d410e57cc38d99f09bb15"},"schema_version":"1.0"},"canonical_sha256":"411ef7a56d1ae04b0883b3f77bc2d607bf2a621926d557b1c07fa50abff88e80","source":{"kind":"arxiv","id":"1502.00016","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.00016","created_at":"2026-05-18T00:26:52Z"},{"alias_kind":"arxiv_version","alias_value":"1502.00016v2","created_at":"2026-05-18T00:26:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.00016","created_at":"2026-05-18T00:26:52Z"},{"alias_kind":"pith_short_12","alias_value":"IEPPPJLNDLQE","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_16","alias_value":"IEPPPJLNDLQEWCED","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_8","alias_value":"IEPPPJLN","created_at":"2026-05-18T12:29:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:IEPPPJLNDLQEWCEDWP3XXQWWA6","target":"record","payload":{"canonical_record":{"source":{"id":"1502.00016","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-30T21:17:27Z","cross_cats_sorted":["quant-ph"],"title_canon_sha256":"831912be0e5029fc768bd6cd7ffafc71702fffe9520bf963c76e4a6ef8b1f4ea","abstract_canon_sha256":"0c36591f8684ef9d9c8c71715ad6cf30333f19be276d410e57cc38d99f09bb15"},"schema_version":"1.0"},"canonical_sha256":"411ef7a56d1ae04b0883b3f77bc2d607bf2a621926d557b1c07fa50abff88e80","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:26:52.227548Z","signature_b64":"tUUsf7LNmGtLMl539YEFSGgMJvkOtPpMbn0tna8gWRuL0da3CQVv/ynZ/Ae/IMm4e6RvJx1tvaolWVUpLjiKCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"411ef7a56d1ae04b0883b3f77bc2d607bf2a621926d557b1c07fa50abff88e80","last_reissued_at":"2026-05-18T00:26:52.226955Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:26:52.226955Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1502.00016","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:26:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"j8tBP2d3jIjQbv6yk94rmMw/JUfSUWhDIf8Nv84eAjyhKBAOP1bNLT1KdcWAZrui49E0WGvUe/PwlW355/n/Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T08:05:55.239955Z"},"content_sha256":"de1d93a2f9845723d69fff4345aed4681f4cd9b62999e10da87efc72b628abfc","schema_version":"1.0","event_id":"sha256:de1d93a2f9845723d69fff4345aed4681f4cd9b62999e10da87efc72b628abfc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:IEPPPJLNDLQEWCEDWP3XXQWWA6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Orthogonal Representations, Projective Rank, and Fractional Minimum Positive Semidefinite Rank: Connections and New Directions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["quant-ph"],"primary_cat":"math.CO","authors_text":"David E. Roberson, Kevin F. Palmowski, Leslie Hogben, Simone Severini","submitted_at":"2015-01-30T21:17:27Z","abstract_excerpt":"Fractional minimum positive semidefinite rank is defined from $r$-fold faithful orthogonal representations and it is shown that the projective rank of any graph equals the fractional minimum positive semidefinite rank of its complement. An $r$-fold version of the traditional definition of minimum positive semidefinite rank of a graph using Hermitian matrices that fit the graph is also presented. This paper also introduces $r$-fold orthogonal representations of graphs and formalizes the understanding of projective rank as fractional orthogonal rank. Connections of these concepts to quantum theo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.00016","kind":"arxiv","version":2},"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":"2026-05-18T00:26:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"v+74WbKFFWSAklfEuySwKmCDirpfWeVfrjk8ZPp1NMQ2YrIUYCVJKIDNC8vtDSRH/ixJrEgDvQi8W6kx3Uh2CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T08:05:55.240705Z"},"content_sha256":"60e5442f55e6eb236178984c15dd2dc50e2de0e90527d22768d7030d2a8ba227","schema_version":"1.0","event_id":"sha256:60e5442f55e6eb236178984c15dd2dc50e2de0e90527d22768d7030d2a8ba227"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IEPPPJLNDLQEWCEDWP3XXQWWA6/bundle.json","state_url":"https://pith.science/pith/IEPPPJLNDLQEWCEDWP3XXQWWA6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IEPPPJLNDLQEWCEDWP3XXQWWA6/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":"2026-06-09T08:05:55Z","links":{"resolver":"https://pith.science/pith/IEPPPJLNDLQEWCEDWP3XXQWWA6","bundle":"https://pith.science/pith/IEPPPJLNDLQEWCEDWP3XXQWWA6/bundle.json","state":"https://pith.science/pith/IEPPPJLNDLQEWCEDWP3XXQWWA6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IEPPPJLNDLQEWCEDWP3XXQWWA6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:IEPPPJLNDLQEWCEDWP3XXQWWA6","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":"0c36591f8684ef9d9c8c71715ad6cf30333f19be276d410e57cc38d99f09bb15","cross_cats_sorted":["quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-30T21:17:27Z","title_canon_sha256":"831912be0e5029fc768bd6cd7ffafc71702fffe9520bf963c76e4a6ef8b1f4ea"},"schema_version":"1.0","source":{"id":"1502.00016","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.00016","created_at":"2026-05-18T00:26:52Z"},{"alias_kind":"arxiv_version","alias_value":"1502.00016v2","created_at":"2026-05-18T00:26:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.00016","created_at":"2026-05-18T00:26:52Z"},{"alias_kind":"pith_short_12","alias_value":"IEPPPJLNDLQE","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_16","alias_value":"IEPPPJLNDLQEWCED","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_8","alias_value":"IEPPPJLN","created_at":"2026-05-18T12:29:25Z"}],"graph_snapshots":[{"event_id":"sha256:60e5442f55e6eb236178984c15dd2dc50e2de0e90527d22768d7030d2a8ba227","target":"graph","created_at":"2026-05-18T00:26:52Z","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":"Fractional minimum positive semidefinite rank is defined from $r$-fold faithful orthogonal representations and it is shown that the projective rank of any graph equals the fractional minimum positive semidefinite rank of its complement. An $r$-fold version of the traditional definition of minimum positive semidefinite rank of a graph using Hermitian matrices that fit the graph is also presented. This paper also introduces $r$-fold orthogonal representations of graphs and formalizes the understanding of projective rank as fractional orthogonal rank. Connections of these concepts to quantum theo","authors_text":"David E. Roberson, Kevin F. Palmowski, Leslie Hogben, Simone Severini","cross_cats":["quant-ph"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-30T21:17:27Z","title":"Orthogonal Representations, Projective Rank, and Fractional Minimum Positive Semidefinite Rank: Connections and New Directions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.00016","kind":"arxiv","version":2},"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:de1d93a2f9845723d69fff4345aed4681f4cd9b62999e10da87efc72b628abfc","target":"record","created_at":"2026-05-18T00:26:52Z","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":"0c36591f8684ef9d9c8c71715ad6cf30333f19be276d410e57cc38d99f09bb15","cross_cats_sorted":["quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-30T21:17:27Z","title_canon_sha256":"831912be0e5029fc768bd6cd7ffafc71702fffe9520bf963c76e4a6ef8b1f4ea"},"schema_version":"1.0","source":{"id":"1502.00016","kind":"arxiv","version":2}},"canonical_sha256":"411ef7a56d1ae04b0883b3f77bc2d607bf2a621926d557b1c07fa50abff88e80","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"411ef7a56d1ae04b0883b3f77bc2d607bf2a621926d557b1c07fa50abff88e80","first_computed_at":"2026-05-18T00:26:52.226955Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:26:52.226955Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tUUsf7LNmGtLMl539YEFSGgMJvkOtPpMbn0tna8gWRuL0da3CQVv/ynZ/Ae/IMm4e6RvJx1tvaolWVUpLjiKCA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:26:52.227548Z","signed_message":"canonical_sha256_bytes"},"source_id":"1502.00016","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:de1d93a2f9845723d69fff4345aed4681f4cd9b62999e10da87efc72b628abfc","sha256:60e5442f55e6eb236178984c15dd2dc50e2de0e90527d22768d7030d2a8ba227"],"state_sha256":"efec0c5350b08c554a11b5e47f8a35b56295a9d3eab7101ee44ed52ef2d99296"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WjxRdvqNvx9DGvT13wCtDQimX54cz1O6jLopdTNn3wJ3KlG/vyOsqgtrW1TWtdr63Fr5Un9ISYAXD+xIXNHlBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T08:05:55.244670Z","bundle_sha256":"cfe8f58fa4fa641d7a4559b548827768f3c9d62cabdfd6f8e01dc40cdead8538"}}