{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:GAJMEGWWPHDDKE3IC74IDYX3JO","short_pith_number":"pith:GAJMEGWW","canonical_record":{"source":{"id":"1004.5166","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-04-29T00:13:26Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"4c5d02b2033c48c99217ab3eac37157275d461cf0fac3764dc3cd17136987dcb","abstract_canon_sha256":"44de40e4d032459e0622c2022951da0feae8ff21d7773c2143a40e4651b6cf6b"},"schema_version":"1.0"},"canonical_sha256":"3012c21ad679c635136817f881e2fb4ba05cb9219f7eaabca4ad68d7d493a6fe","source":{"kind":"arxiv","id":"1004.5166","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.5166","created_at":"2026-05-18T04:27:15Z"},{"alias_kind":"arxiv_version","alias_value":"1004.5166v2","created_at":"2026-05-18T04:27:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.5166","created_at":"2026-05-18T04:27:15Z"},{"alias_kind":"pith_short_12","alias_value":"GAJMEGWWPHDD","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"GAJMEGWWPHDDKE3I","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"GAJMEGWW","created_at":"2026-05-18T12:26:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:GAJMEGWWPHDDKE3IC74IDYX3JO","target":"record","payload":{"canonical_record":{"source":{"id":"1004.5166","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-04-29T00:13:26Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"4c5d02b2033c48c99217ab3eac37157275d461cf0fac3764dc3cd17136987dcb","abstract_canon_sha256":"44de40e4d032459e0622c2022951da0feae8ff21d7773c2143a40e4651b6cf6b"},"schema_version":"1.0"},"canonical_sha256":"3012c21ad679c635136817f881e2fb4ba05cb9219f7eaabca4ad68d7d493a6fe","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:27:15.030689Z","signature_b64":"uNa0Z3eW21+1tDeHqNd0min6jEtO2/nruSioOnNxyCuovn7Qzah2bGCCwc37utesIKe5m9ui9ABnHLuvhVNGDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3012c21ad679c635136817f881e2fb4ba05cb9219f7eaabca4ad68d7d493a6fe","last_reissued_at":"2026-05-18T04:27:15.030036Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:27:15.030036Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1004.5166","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":"2026-05-18T04:27:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"doEjrKQoh7SRVXoj6kCkG0mGIyxC1cXqzClR7s/4Frp+d+y2UJf+VlsDaltsFSNxpXhzMBFW7qMzEcVQ0aBUAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T20:01:30.170595Z"},"content_sha256":"7ae5726d9c1d9eca693d42cb07aa3771950a8ae5bf1ccc6a41340d8f86b5d40e","schema_version":"1.0","event_id":"sha256:7ae5726d9c1d9eca693d42cb07aa3771950a8ae5bf1ccc6a41340d8f86b5d40e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:GAJMEGWWPHDDKE3IC74IDYX3JO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Singular Structure of Graph Hypersurfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AG","authors_text":"Eric Patterson","submitted_at":"2010-04-29T00:13:26Z","abstract_excerpt":"We show that the singular loci of graph hypersurfaces correspond set-theoretically to their rank loci. The proof holds for all configuration hypersurfaces and depends only on linear algebra. To make the conclusion for the second graph hypersurface, we prove that the second graph polynomial is a configuration polynomial. The result indicates that there may be a fruitful interplay between the current research in graph hypersurfaces and Stratified Morse Theory."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.5166","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":"2026-05-18T04:27:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nwzEOpTnrMduhM1mdHfK+BO3thazDAHfcuC/P7xr8Eh4GwLe07WYdTwUQKuVi7RJNePpoE5Rr+7hm93Qk4HPCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T20:01:30.170942Z"},"content_sha256":"57a75614d2f31d35bdc9ca2079e0aa4029df9da0edefe60bdea50b7ae16e4371","schema_version":"1.0","event_id":"sha256:57a75614d2f31d35bdc9ca2079e0aa4029df9da0edefe60bdea50b7ae16e4371"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GAJMEGWWPHDDKE3IC74IDYX3JO/bundle.json","state_url":"https://pith.science/pith/GAJMEGWWPHDDKE3IC74IDYX3JO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GAJMEGWWPHDDKE3IC74IDYX3JO/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":"2026-06-25T20:01:30Z","links":{"resolver":"https://pith.science/pith/GAJMEGWWPHDDKE3IC74IDYX3JO","bundle":"https://pith.science/pith/GAJMEGWWPHDDKE3IC74IDYX3JO/bundle.json","state":"https://pith.science/pith/GAJMEGWWPHDDKE3IC74IDYX3JO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GAJMEGWWPHDDKE3IC74IDYX3JO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:GAJMEGWWPHDDKE3IC74IDYX3JO","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":"44de40e4d032459e0622c2022951da0feae8ff21d7773c2143a40e4651b6cf6b","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-04-29T00:13:26Z","title_canon_sha256":"4c5d02b2033c48c99217ab3eac37157275d461cf0fac3764dc3cd17136987dcb"},"schema_version":"1.0","source":{"id":"1004.5166","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.5166","created_at":"2026-05-18T04:27:15Z"},{"alias_kind":"arxiv_version","alias_value":"1004.5166v2","created_at":"2026-05-18T04:27:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.5166","created_at":"2026-05-18T04:27:15Z"},{"alias_kind":"pith_short_12","alias_value":"GAJMEGWWPHDD","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"GAJMEGWWPHDDKE3I","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"GAJMEGWW","created_at":"2026-05-18T12:26:07Z"}],"graph_snapshots":[{"event_id":"sha256:57a75614d2f31d35bdc9ca2079e0aa4029df9da0edefe60bdea50b7ae16e4371","target":"graph","created_at":"2026-05-18T04:27:15Z","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 show that the singular loci of graph hypersurfaces correspond set-theoretically to their rank loci. The proof holds for all configuration hypersurfaces and depends only on linear algebra. To make the conclusion for the second graph hypersurface, we prove that the second graph polynomial is a configuration polynomial. The result indicates that there may be a fruitful interplay between the current research in graph hypersurfaces and Stratified Morse Theory.","authors_text":"Eric Patterson","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-04-29T00:13:26Z","title":"On the Singular Structure of Graph Hypersurfaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.5166","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":"sha256:7ae5726d9c1d9eca693d42cb07aa3771950a8ae5bf1ccc6a41340d8f86b5d40e","target":"record","created_at":"2026-05-18T04:27:15Z","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":"44de40e4d032459e0622c2022951da0feae8ff21d7773c2143a40e4651b6cf6b","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-04-29T00:13:26Z","title_canon_sha256":"4c5d02b2033c48c99217ab3eac37157275d461cf0fac3764dc3cd17136987dcb"},"schema_version":"1.0","source":{"id":"1004.5166","kind":"arxiv","version":2}},"canonical_sha256":"3012c21ad679c635136817f881e2fb4ba05cb9219f7eaabca4ad68d7d493a6fe","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3012c21ad679c635136817f881e2fb4ba05cb9219f7eaabca4ad68d7d493a6fe","first_computed_at":"2026-05-18T04:27:15.030036Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:27:15.030036Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uNa0Z3eW21+1tDeHqNd0min6jEtO2/nruSioOnNxyCuovn7Qzah2bGCCwc37utesIKe5m9ui9ABnHLuvhVNGDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:27:15.030689Z","signed_message":"canonical_sha256_bytes"},"source_id":"1004.5166","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7ae5726d9c1d9eca693d42cb07aa3771950a8ae5bf1ccc6a41340d8f86b5d40e","sha256:57a75614d2f31d35bdc9ca2079e0aa4029df9da0edefe60bdea50b7ae16e4371"],"state_sha256":"b7968f3a3032f1913d7ade0746f7e0dc61be6f033da001651b0fe454dc1df562"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oGODlRMNbGaINyqwK+jRA1835FqWaDM8I8PrTkV6BesA3IFEY7K4VftoLFK4JGg+B48elAhVvE21qlMELnGQDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T20:01:30.172830Z","bundle_sha256":"aac8a533f667bd13c6d1373aae27d40d31a560a2a4d13ef8743ff25bb9b1d451"}}