{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:BNPJBPKUH2UJS3QUXNIYIU3FRG","short_pith_number":"pith:BNPJBPKU","canonical_record":{"source":{"id":"1111.6007","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-11-25T14:47:45Z","cross_cats_sorted":[],"title_canon_sha256":"11093fc64c8dedcee029ba367d19133be1f36ed84863fab15b3b94b3c8e3a961","abstract_canon_sha256":"c850c7be9fd359b08d06126155405f87f06a12d1055439a7c51f11e59e3f187f"},"schema_version":"1.0"},"canonical_sha256":"0b5e90bd543ea8996e14bb51845365899b4a94a26aa6165bcf97509ff3787304","source":{"kind":"arxiv","id":"1111.6007","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.6007","created_at":"2026-05-18T04:07:41Z"},{"alias_kind":"arxiv_version","alias_value":"1111.6007v1","created_at":"2026-05-18T04:07:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.6007","created_at":"2026-05-18T04:07:41Z"},{"alias_kind":"pith_short_12","alias_value":"BNPJBPKUH2UJ","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"BNPJBPKUH2UJS3QU","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"BNPJBPKU","created_at":"2026-05-18T12:26:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:BNPJBPKUH2UJS3QUXNIYIU3FRG","target":"record","payload":{"canonical_record":{"source":{"id":"1111.6007","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-11-25T14:47:45Z","cross_cats_sorted":[],"title_canon_sha256":"11093fc64c8dedcee029ba367d19133be1f36ed84863fab15b3b94b3c8e3a961","abstract_canon_sha256":"c850c7be9fd359b08d06126155405f87f06a12d1055439a7c51f11e59e3f187f"},"schema_version":"1.0"},"canonical_sha256":"0b5e90bd543ea8996e14bb51845365899b4a94a26aa6165bcf97509ff3787304","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:07:41.271698Z","signature_b64":"Y0q1IYFwO88jcgiQo/yPAa5MBvrlm2lgXadrupn2LM4nudOVUXl4eHg0PvuXS9Ssc8oh8+Edgp/2tJ7t7guYAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0b5e90bd543ea8996e14bb51845365899b4a94a26aa6165bcf97509ff3787304","last_reissued_at":"2026-05-18T04:07:41.270964Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:07:41.270964Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1111.6007","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":"2026-05-18T04:07:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"skl06UtgLQHAvVxcIZsH8YfiyCA4PdCTJDDqubQDahnYpUg+Nm0xbxHNq9vSDK/k931jUaIly4NPbCf47oqeBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T11:48:45.358144Z"},"content_sha256":"3a1a6517a9151e667e6fde26485cd409e76ece55d7b1bac7e82921ed8ec8c747","schema_version":"1.0","event_id":"sha256:3a1a6517a9151e667e6fde26485cd409e76ece55d7b1bac7e82921ed8ec8c747"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:BNPJBPKUH2UJS3QUXNIYIU3FRG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the density of triangles and squares in regular finite and unimodular random graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Viktor Harangi","submitted_at":"2011-11-25T14:47:45Z","abstract_excerpt":"We explicitly describe the possible pairs of triangle and square densities for r-regular finite simple graphs. We also prove that every r-regular unimodular random graph can be approximated by r-regular finite graphs with respect to these densities. As a corollary one gets an explicit description of the possible pairs of the third and fourth moments of the spectral measure of r-regular unimodular random graphs."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.6007","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":"2026-05-18T04:07:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"llpdghokTBh6xFm77D+FLIrxHh1xdTCmjZvxZzjtlc+2sR/iNyi6IBDl5YOGhFmnwHguSeCi5PocsSIkiKI7BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T11:48:45.358704Z"},"content_sha256":"dc26987e06782bfb8166f2fe41b9c899e6862d9b070e3f1be0a111610526b084","schema_version":"1.0","event_id":"sha256:dc26987e06782bfb8166f2fe41b9c899e6862d9b070e3f1be0a111610526b084"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BNPJBPKUH2UJS3QUXNIYIU3FRG/bundle.json","state_url":"https://pith.science/pith/BNPJBPKUH2UJS3QUXNIYIU3FRG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BNPJBPKUH2UJS3QUXNIYIU3FRG/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":"2026-06-07T11:48:45Z","links":{"resolver":"https://pith.science/pith/BNPJBPKUH2UJS3QUXNIYIU3FRG","bundle":"https://pith.science/pith/BNPJBPKUH2UJS3QUXNIYIU3FRG/bundle.json","state":"https://pith.science/pith/BNPJBPKUH2UJS3QUXNIYIU3FRG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BNPJBPKUH2UJS3QUXNIYIU3FRG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:BNPJBPKUH2UJS3QUXNIYIU3FRG","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":"c850c7be9fd359b08d06126155405f87f06a12d1055439a7c51f11e59e3f187f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-11-25T14:47:45Z","title_canon_sha256":"11093fc64c8dedcee029ba367d19133be1f36ed84863fab15b3b94b3c8e3a961"},"schema_version":"1.0","source":{"id":"1111.6007","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.6007","created_at":"2026-05-18T04:07:41Z"},{"alias_kind":"arxiv_version","alias_value":"1111.6007v1","created_at":"2026-05-18T04:07:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.6007","created_at":"2026-05-18T04:07:41Z"},{"alias_kind":"pith_short_12","alias_value":"BNPJBPKUH2UJ","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"BNPJBPKUH2UJS3QU","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"BNPJBPKU","created_at":"2026-05-18T12:26:24Z"}],"graph_snapshots":[{"event_id":"sha256:dc26987e06782bfb8166f2fe41b9c899e6862d9b070e3f1be0a111610526b084","target":"graph","created_at":"2026-05-18T04:07:41Z","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":"We explicitly describe the possible pairs of triangle and square densities for r-regular finite simple graphs. We also prove that every r-regular unimodular random graph can be approximated by r-regular finite graphs with respect to these densities. As a corollary one gets an explicit description of the possible pairs of the third and fourth moments of the spectral measure of r-regular unimodular random graphs.","authors_text":"Viktor Harangi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-11-25T14:47:45Z","title":"On the density of triangles and squares in regular finite and unimodular random graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.6007","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":"sha256:3a1a6517a9151e667e6fde26485cd409e76ece55d7b1bac7e82921ed8ec8c747","target":"record","created_at":"2026-05-18T04:07:41Z","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":"c850c7be9fd359b08d06126155405f87f06a12d1055439a7c51f11e59e3f187f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-11-25T14:47:45Z","title_canon_sha256":"11093fc64c8dedcee029ba367d19133be1f36ed84863fab15b3b94b3c8e3a961"},"schema_version":"1.0","source":{"id":"1111.6007","kind":"arxiv","version":1}},"canonical_sha256":"0b5e90bd543ea8996e14bb51845365899b4a94a26aa6165bcf97509ff3787304","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0b5e90bd543ea8996e14bb51845365899b4a94a26aa6165bcf97509ff3787304","first_computed_at":"2026-05-18T04:07:41.270964Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:07:41.270964Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Y0q1IYFwO88jcgiQo/yPAa5MBvrlm2lgXadrupn2LM4nudOVUXl4eHg0PvuXS9Ssc8oh8+Edgp/2tJ7t7guYAw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:07:41.271698Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.6007","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3a1a6517a9151e667e6fde26485cd409e76ece55d7b1bac7e82921ed8ec8c747","sha256:dc26987e06782bfb8166f2fe41b9c899e6862d9b070e3f1be0a111610526b084"],"state_sha256":"846fe77647d9ebfb98897f39c0228eedbe44ac9c791e7408e0ab03e83083080e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FzE4Ro7575jVodfhOvxA7gX49XwyCMv2XV0/ozj5WVq/sXG+qDcvFX6DpRNJAtxCUa8NwP6fRGIJ3DxSSQNcDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T11:48:45.361435Z","bundle_sha256":"aad5164f1c041aae57ebd9ecadaf480209ea450e0465d8f9114c804da9e807ca"}}