{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:CFQFQT2J5MUUZTHQW5YXZ4ZGHE","short_pith_number":"pith:CFQFQT2J","canonical_record":{"source":{"id":"1612.09554","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-30T18:37:10Z","cross_cats_sorted":[],"title_canon_sha256":"a790ee6dd2bed715b9dc3359967b1a2ffd8f2a31764e59804278280e2e12eb30","abstract_canon_sha256":"f2a336aa6acf0e8ede051a20436d2eda0445f3c00c83aadf61e526a893663e95"},"schema_version":"1.0"},"canonical_sha256":"1160584f49eb294cccf0b7717cf32639243d2a4f79ba7eb8f51a6f0447e479cf","source":{"kind":"arxiv","id":"1612.09554","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.09554","created_at":"2026-05-18T00:53:41Z"},{"alias_kind":"arxiv_version","alias_value":"1612.09554v1","created_at":"2026-05-18T00:53:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.09554","created_at":"2026-05-18T00:53:41Z"},{"alias_kind":"pith_short_12","alias_value":"CFQFQT2J5MUU","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_16","alias_value":"CFQFQT2J5MUUZTHQ","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_8","alias_value":"CFQFQT2J","created_at":"2026-05-18T12:30:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:CFQFQT2J5MUUZTHQW5YXZ4ZGHE","target":"record","payload":{"canonical_record":{"source":{"id":"1612.09554","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-30T18:37:10Z","cross_cats_sorted":[],"title_canon_sha256":"a790ee6dd2bed715b9dc3359967b1a2ffd8f2a31764e59804278280e2e12eb30","abstract_canon_sha256":"f2a336aa6acf0e8ede051a20436d2eda0445f3c00c83aadf61e526a893663e95"},"schema_version":"1.0"},"canonical_sha256":"1160584f49eb294cccf0b7717cf32639243d2a4f79ba7eb8f51a6f0447e479cf","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:41.232638Z","signature_b64":"A4GWuzJZqlcjHhu/MtqJ3WZN1KCexYY9Kh/Cqn9TCNI/IFuifGZvRwVizWbDmZ6uAIZSNa0AmPx+Eg5uijP8Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1160584f49eb294cccf0b7717cf32639243d2a4f79ba7eb8f51a6f0447e479cf","last_reissued_at":"2026-05-18T00:53:41.232161Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:41.232161Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1612.09554","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":"2026-05-18T00:53:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MzjRKRX4/iKMVErkWMR4m5l7AzhY/4YW4UaTsEipP+3+jgvFJRDlVe1qmZzhllR1cz99eVuCGER/VLNv/eumBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T02:55:32.321848Z"},"content_sha256":"86aa43455b940338d3a20acff009e3a76a11cbadaea754134ef106fdbd79f7df","schema_version":"1.0","event_id":"sha256:86aa43455b940338d3a20acff009e3a76a11cbadaea754134ef106fdbd79f7df"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:CFQFQT2J5MUUZTHQW5YXZ4ZGHE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the boundary of the region defined by homomorphism densities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hamed Hatami, Sergey Norin","submitted_at":"2016-12-30T18:37:10Z","abstract_excerpt":"The Kruskal-Katona theorem together with a theorem of Razborov determine the closure of the set of points defined by the homomorphism density of the edge and the triangle in finite graphs. The boundary of this region is a countable union of algebraic curves, and in particular, it is almost everywhere differentiable. One can more generally consider the region defined by the homomorphism densities of a list of given graphs, and ask whether the boundary is as well-behaved as in the case of the triangle and the edge. Towards answering this question in the negative, we construct examples which show"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.09554","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":"2026-05-18T00:53:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dlIRFbDMzsHvi8GExTZONC3tJpxplH8xNRR+iyuzggpTTLgvLtQQMCxVJSDIKyh57wUMfAKkrmr53f9NU2tJCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T02:55:32.322498Z"},"content_sha256":"a34ce33c8908611b44792f47f55c3fee75ae9e8536900870185a281ffc769c04","schema_version":"1.0","event_id":"sha256:a34ce33c8908611b44792f47f55c3fee75ae9e8536900870185a281ffc769c04"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CFQFQT2J5MUUZTHQW5YXZ4ZGHE/bundle.json","state_url":"https://pith.science/pith/CFQFQT2J5MUUZTHQW5YXZ4ZGHE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CFQFQT2J5MUUZTHQW5YXZ4ZGHE/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":"2026-05-26T02:55:32Z","links":{"resolver":"https://pith.science/pith/CFQFQT2J5MUUZTHQW5YXZ4ZGHE","bundle":"https://pith.science/pith/CFQFQT2J5MUUZTHQW5YXZ4ZGHE/bundle.json","state":"https://pith.science/pith/CFQFQT2J5MUUZTHQW5YXZ4ZGHE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CFQFQT2J5MUUZTHQW5YXZ4ZGHE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:CFQFQT2J5MUUZTHQW5YXZ4ZGHE","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":"f2a336aa6acf0e8ede051a20436d2eda0445f3c00c83aadf61e526a893663e95","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-30T18:37:10Z","title_canon_sha256":"a790ee6dd2bed715b9dc3359967b1a2ffd8f2a31764e59804278280e2e12eb30"},"schema_version":"1.0","source":{"id":"1612.09554","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.09554","created_at":"2026-05-18T00:53:41Z"},{"alias_kind":"arxiv_version","alias_value":"1612.09554v1","created_at":"2026-05-18T00:53:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.09554","created_at":"2026-05-18T00:53:41Z"},{"alias_kind":"pith_short_12","alias_value":"CFQFQT2J5MUU","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_16","alias_value":"CFQFQT2J5MUUZTHQ","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_8","alias_value":"CFQFQT2J","created_at":"2026-05-18T12:30:09Z"}],"graph_snapshots":[{"event_id":"sha256:a34ce33c8908611b44792f47f55c3fee75ae9e8536900870185a281ffc769c04","target":"graph","created_at":"2026-05-18T00:53:41Z","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":"The Kruskal-Katona theorem together with a theorem of Razborov determine the closure of the set of points defined by the homomorphism density of the edge and the triangle in finite graphs. The boundary of this region is a countable union of algebraic curves, and in particular, it is almost everywhere differentiable. One can more generally consider the region defined by the homomorphism densities of a list of given graphs, and ask whether the boundary is as well-behaved as in the case of the triangle and the edge. Towards answering this question in the negative, we construct examples which show","authors_text":"Hamed Hatami, Sergey Norin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-30T18:37:10Z","title":"On the boundary of the region defined by homomorphism densities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.09554","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":"sha256:86aa43455b940338d3a20acff009e3a76a11cbadaea754134ef106fdbd79f7df","target":"record","created_at":"2026-05-18T00:53:41Z","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":"f2a336aa6acf0e8ede051a20436d2eda0445f3c00c83aadf61e526a893663e95","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-30T18:37:10Z","title_canon_sha256":"a790ee6dd2bed715b9dc3359967b1a2ffd8f2a31764e59804278280e2e12eb30"},"schema_version":"1.0","source":{"id":"1612.09554","kind":"arxiv","version":1}},"canonical_sha256":"1160584f49eb294cccf0b7717cf32639243d2a4f79ba7eb8f51a6f0447e479cf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1160584f49eb294cccf0b7717cf32639243d2a4f79ba7eb8f51a6f0447e479cf","first_computed_at":"2026-05-18T00:53:41.232161Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:53:41.232161Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"A4GWuzJZqlcjHhu/MtqJ3WZN1KCexYY9Kh/Cqn9TCNI/IFuifGZvRwVizWbDmZ6uAIZSNa0AmPx+Eg5uijP8Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:53:41.232638Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.09554","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:86aa43455b940338d3a20acff009e3a76a11cbadaea754134ef106fdbd79f7df","sha256:a34ce33c8908611b44792f47f55c3fee75ae9e8536900870185a281ffc769c04"],"state_sha256":"29a2e7890bfb5f8bc9c148486a3265a4d904f3ba1e318d4cf2d73f5a1b48fb5c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"crw8aa1aqKJODjrIHXE5flPyZO3Vjue8D2xDVAiPVlNpQDeAJQDy0fIERWC3WeWFwTDRp3GHSHNUxP0kKn84BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T02:55:32.326411Z","bundle_sha256":"2a86ea2f7b119b9266e8a8bc90d28f18ebe5b0f742cae97322b29d49cfa2cf34"}}