{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:4R76QCNLEIMU2F2XRHAVZGZMOD","short_pith_number":"pith:4R76QCNL","canonical_record":{"source":{"id":"1601.03981","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-15T16:06:27Z","cross_cats_sorted":[],"title_canon_sha256":"30e69ba2416773eb731f182951dd0d9494e84f76f40ea5e4e057d4667b148fd0","abstract_canon_sha256":"abf56f67e91f2c06161bb1f00efc7d4da4daad82c4e40d38bde0f9c698e05b7c"},"schema_version":"1.0"},"canonical_sha256":"e47fe809ab22194d175789c15c9b2c70e292efebee83c3007ce23f2779a524fd","source":{"kind":"arxiv","id":"1601.03981","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.03981","created_at":"2026-05-18T00:58:34Z"},{"alias_kind":"arxiv_version","alias_value":"1601.03981v3","created_at":"2026-05-18T00:58:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.03981","created_at":"2026-05-18T00:58:34Z"},{"alias_kind":"pith_short_12","alias_value":"4R76QCNLEIMU","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_16","alias_value":"4R76QCNLEIMU2F2X","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_8","alias_value":"4R76QCNL","created_at":"2026-05-18T12:29:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:4R76QCNLEIMU2F2XRHAVZGZMOD","target":"record","payload":{"canonical_record":{"source":{"id":"1601.03981","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-15T16:06:27Z","cross_cats_sorted":[],"title_canon_sha256":"30e69ba2416773eb731f182951dd0d9494e84f76f40ea5e4e057d4667b148fd0","abstract_canon_sha256":"abf56f67e91f2c06161bb1f00efc7d4da4daad82c4e40d38bde0f9c698e05b7c"},"schema_version":"1.0"},"canonical_sha256":"e47fe809ab22194d175789c15c9b2c70e292efebee83c3007ce23f2779a524fd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:58:34.279536Z","signature_b64":"0NWBvWM9W1C6LQq0PyOZWMdcQGkAORJLAFzSgKcYFhPyKCO/Owk2/tWfAu8ZZxdyedxyocEdQLg+iJgh5veXAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e47fe809ab22194d175789c15c9b2c70e292efebee83c3007ce23f2779a524fd","last_reissued_at":"2026-05-18T00:58:34.278906Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:58:34.278906Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1601.03981","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":"2026-05-18T00:58:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qZJmM2dpQpRWZ5IbHu1ZNNti+6lp98wJFhgznINIbkmElYFt+R7ekaj58cuTItdnwxD5a6Bp7EQwZBxuv7tDCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T03:19:32.960716Z"},"content_sha256":"c68652ecbd4f14254c61d0da52ac37027de9572290b33b44f5ab653fe1afa94b","schema_version":"1.0","event_id":"sha256:c68652ecbd4f14254c61d0da52ac37027de9572290b33b44f5ab653fe1afa94b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:4R76QCNLEIMU2F2XRHAVZGZMOD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A point-line incidence identity in finite fields, and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Brendan Murphy, Giorgis Petridis","submitted_at":"2016-01-15T16:06:27Z","abstract_excerpt":"Let $E \\subseteq \\mathbb{F}_q^2$ be a set in the 2-dimensional vector space over a finite field with $q$ elements. We prove an identity for the second moment of its incidence function and deduce a variety of existing results from the literature, not all naturally associated with lines in $\\mathbb{F}_q^2$, in a unified and elementary way."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.03981","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":"2026-05-18T00:58:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mlt2iCaITFh0eXT2H3OHvcoSMeTmg9z9Xs2NS20sxqEZQWLMudJjxL7lbpnHCH4LOxyDA8278yqVMvNOF9IiAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T03:19:32.961387Z"},"content_sha256":"373c041877a227b310ee1d011c29c6de651c506cf363f25776b98921763295d4","schema_version":"1.0","event_id":"sha256:373c041877a227b310ee1d011c29c6de651c506cf363f25776b98921763295d4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4R76QCNLEIMU2F2XRHAVZGZMOD/bundle.json","state_url":"https://pith.science/pith/4R76QCNLEIMU2F2XRHAVZGZMOD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4R76QCNLEIMU2F2XRHAVZGZMOD/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":"2026-06-26T03:19:32Z","links":{"resolver":"https://pith.science/pith/4R76QCNLEIMU2F2XRHAVZGZMOD","bundle":"https://pith.science/pith/4R76QCNLEIMU2F2XRHAVZGZMOD/bundle.json","state":"https://pith.science/pith/4R76QCNLEIMU2F2XRHAVZGZMOD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4R76QCNLEIMU2F2XRHAVZGZMOD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:4R76QCNLEIMU2F2XRHAVZGZMOD","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":"abf56f67e91f2c06161bb1f00efc7d4da4daad82c4e40d38bde0f9c698e05b7c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-15T16:06:27Z","title_canon_sha256":"30e69ba2416773eb731f182951dd0d9494e84f76f40ea5e4e057d4667b148fd0"},"schema_version":"1.0","source":{"id":"1601.03981","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.03981","created_at":"2026-05-18T00:58:34Z"},{"alias_kind":"arxiv_version","alias_value":"1601.03981v3","created_at":"2026-05-18T00:58:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.03981","created_at":"2026-05-18T00:58:34Z"},{"alias_kind":"pith_short_12","alias_value":"4R76QCNLEIMU","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_16","alias_value":"4R76QCNLEIMU2F2X","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_8","alias_value":"4R76QCNL","created_at":"2026-05-18T12:29:58Z"}],"graph_snapshots":[{"event_id":"sha256:373c041877a227b310ee1d011c29c6de651c506cf363f25776b98921763295d4","target":"graph","created_at":"2026-05-18T00:58:34Z","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":"Let $E \\subseteq \\mathbb{F}_q^2$ be a set in the 2-dimensional vector space over a finite field with $q$ elements. We prove an identity for the second moment of its incidence function and deduce a variety of existing results from the literature, not all naturally associated with lines in $\\mathbb{F}_q^2$, in a unified and elementary way.","authors_text":"Brendan Murphy, Giorgis Petridis","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-15T16:06:27Z","title":"A point-line incidence identity in finite fields, and applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.03981","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:c68652ecbd4f14254c61d0da52ac37027de9572290b33b44f5ab653fe1afa94b","target":"record","created_at":"2026-05-18T00:58:34Z","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":"abf56f67e91f2c06161bb1f00efc7d4da4daad82c4e40d38bde0f9c698e05b7c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-15T16:06:27Z","title_canon_sha256":"30e69ba2416773eb731f182951dd0d9494e84f76f40ea5e4e057d4667b148fd0"},"schema_version":"1.0","source":{"id":"1601.03981","kind":"arxiv","version":3}},"canonical_sha256":"e47fe809ab22194d175789c15c9b2c70e292efebee83c3007ce23f2779a524fd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e47fe809ab22194d175789c15c9b2c70e292efebee83c3007ce23f2779a524fd","first_computed_at":"2026-05-18T00:58:34.278906Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:58:34.278906Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0NWBvWM9W1C6LQq0PyOZWMdcQGkAORJLAFzSgKcYFhPyKCO/Owk2/tWfAu8ZZxdyedxyocEdQLg+iJgh5veXAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:58:34.279536Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.03981","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c68652ecbd4f14254c61d0da52ac37027de9572290b33b44f5ab653fe1afa94b","sha256:373c041877a227b310ee1d011c29c6de651c506cf363f25776b98921763295d4"],"state_sha256":"9113cdda8ad5405b60e795ca83641ee8ff377f59b0732614815a36b0f94d4003"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bmydPwk3iKwBg0TnoZt8gel2g0hEWgjHwP6/YRx4i/zIR1wzhI5hUrMrcGtfUGrZLT1X4ieDsr6DtToKYsDXCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T03:19:32.964151Z","bundle_sha256":"f8a332b048b9e7b28f1be21ec0ea7bb2fba75f8302551cc5248d072ff6a04a1b"}}