{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:4I2USVUCFC4GY7UO5TWIOGNCZ4","short_pith_number":"pith:4I2USVUC","canonical_record":{"source":{"id":"1508.05142","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-08-20T22:50:17Z","cross_cats_sorted":[],"title_canon_sha256":"da310320a39e99e66295699b9db2af7b8cfa9951aae01a51c7ce8f18769bef09","abstract_canon_sha256":"14e2c3cd792e9cd93e304f6775affb262e1bd3c233bd49aff2844dfc45f1c029"},"schema_version":"1.0"},"canonical_sha256":"e23549568228b86c7e8eecec8719a2cf39f143006a7254d37b9ba3b482bea467","source":{"kind":"arxiv","id":"1508.05142","version":6},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.05142","created_at":"2026-05-18T01:23:12Z"},{"alias_kind":"arxiv_version","alias_value":"1508.05142v6","created_at":"2026-05-18T01:23:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.05142","created_at":"2026-05-18T01:23:12Z"},{"alias_kind":"pith_short_12","alias_value":"4I2USVUCFC4G","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_16","alias_value":"4I2USVUCFC4GY7UO","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_8","alias_value":"4I2USVUC","created_at":"2026-05-18T12:29:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:4I2USVUCFC4GY7UO5TWIOGNCZ4","target":"record","payload":{"canonical_record":{"source":{"id":"1508.05142","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-08-20T22:50:17Z","cross_cats_sorted":[],"title_canon_sha256":"da310320a39e99e66295699b9db2af7b8cfa9951aae01a51c7ce8f18769bef09","abstract_canon_sha256":"14e2c3cd792e9cd93e304f6775affb262e1bd3c233bd49aff2844dfc45f1c029"},"schema_version":"1.0"},"canonical_sha256":"e23549568228b86c7e8eecec8719a2cf39f143006a7254d37b9ba3b482bea467","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:23:12.023034Z","signature_b64":"JAk9wYGGTwxx+Tzb7G7ai4Ph9s+/imiDGaDj5HNVyBTy1YPoQ7JC7Nj7OtjYgPMcvk1LP+mbMCgYNfiDl+uVDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e23549568228b86c7e8eecec8719a2cf39f143006a7254d37b9ba3b482bea467","last_reissued_at":"2026-05-18T01:23:12.022567Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:23:12.022567Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1508.05142","source_version":6,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:23:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CN/B1Lb7Jgts/HL7WrB8QVelHp5k2Ww1CI+7HAxzRcQrqdqlpKLjLPGMNjvFut4rDOhpNJx/Z5tfoLcvP7sOBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T08:28:34.862503Z"},"content_sha256":"5c456db5c45ed2792ee725e855800b18f3c6b030b2419e7283840630a36ff114","schema_version":"1.0","event_id":"sha256:5c456db5c45ed2792ee725e855800b18f3c6b030b2419e7283840630a36ff114"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:4I2USVUCFC4GY7UO5TWIOGNCZ4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On distinct cross-ratios in positive characteristic","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Misha Rudnev","submitted_at":"2015-08-20T22:50:17Z","abstract_excerpt":"We prove that a finite subset $A$ of the projective line $FP^1$ over a field $F$ of positive characteristic $p$ determines $\\Omega(|A|^2)$ distinct cross-ratios, as long as $|A|<\\sqrt{p}.$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.05142","kind":"arxiv","version":6},"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":"2026-05-18T01:23:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mWIez2zjZycZ4fEj4ley2ryQtCoRv9Nmb9u/MtpWCiJ+lIc/S3axZYcMunNKkH+SBpgVOtKwwpw/Ma3xibYFBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T08:28:34.862844Z"},"content_sha256":"b32d88a74c572b0b1e3100161dcb07e5841c9fbf698b91765d2a1225143c65bc","schema_version":"1.0","event_id":"sha256:b32d88a74c572b0b1e3100161dcb07e5841c9fbf698b91765d2a1225143c65bc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4I2USVUCFC4GY7UO5TWIOGNCZ4/bundle.json","state_url":"https://pith.science/pith/4I2USVUCFC4GY7UO5TWIOGNCZ4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4I2USVUCFC4GY7UO5TWIOGNCZ4/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":"2026-06-02T08:28:34Z","links":{"resolver":"https://pith.science/pith/4I2USVUCFC4GY7UO5TWIOGNCZ4","bundle":"https://pith.science/pith/4I2USVUCFC4GY7UO5TWIOGNCZ4/bundle.json","state":"https://pith.science/pith/4I2USVUCFC4GY7UO5TWIOGNCZ4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4I2USVUCFC4GY7UO5TWIOGNCZ4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:4I2USVUCFC4GY7UO5TWIOGNCZ4","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":"14e2c3cd792e9cd93e304f6775affb262e1bd3c233bd49aff2844dfc45f1c029","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-08-20T22:50:17Z","title_canon_sha256":"da310320a39e99e66295699b9db2af7b8cfa9951aae01a51c7ce8f18769bef09"},"schema_version":"1.0","source":{"id":"1508.05142","kind":"arxiv","version":6}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.05142","created_at":"2026-05-18T01:23:12Z"},{"alias_kind":"arxiv_version","alias_value":"1508.05142v6","created_at":"2026-05-18T01:23:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.05142","created_at":"2026-05-18T01:23:12Z"},{"alias_kind":"pith_short_12","alias_value":"4I2USVUCFC4G","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_16","alias_value":"4I2USVUCFC4GY7UO","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_8","alias_value":"4I2USVUC","created_at":"2026-05-18T12:29:05Z"}],"graph_snapshots":[{"event_id":"sha256:b32d88a74c572b0b1e3100161dcb07e5841c9fbf698b91765d2a1225143c65bc","target":"graph","created_at":"2026-05-18T01:23:12Z","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 prove that a finite subset $A$ of the projective line $FP^1$ over a field $F$ of positive characteristic $p$ determines $\\Omega(|A|^2)$ distinct cross-ratios, as long as $|A|<\\sqrt{p}.$","authors_text":"Misha Rudnev","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-08-20T22:50:17Z","title":"On distinct cross-ratios in positive characteristic"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.05142","kind":"arxiv","version":6},"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":"sha256:5c456db5c45ed2792ee725e855800b18f3c6b030b2419e7283840630a36ff114","target":"record","created_at":"2026-05-18T01:23:12Z","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":"14e2c3cd792e9cd93e304f6775affb262e1bd3c233bd49aff2844dfc45f1c029","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-08-20T22:50:17Z","title_canon_sha256":"da310320a39e99e66295699b9db2af7b8cfa9951aae01a51c7ce8f18769bef09"},"schema_version":"1.0","source":{"id":"1508.05142","kind":"arxiv","version":6}},"canonical_sha256":"e23549568228b86c7e8eecec8719a2cf39f143006a7254d37b9ba3b482bea467","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e23549568228b86c7e8eecec8719a2cf39f143006a7254d37b9ba3b482bea467","first_computed_at":"2026-05-18T01:23:12.022567Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:23:12.022567Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JAk9wYGGTwxx+Tzb7G7ai4Ph9s+/imiDGaDj5HNVyBTy1YPoQ7JC7Nj7OtjYgPMcvk1LP+mbMCgYNfiDl+uVDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:23:12.023034Z","signed_message":"canonical_sha256_bytes"},"source_id":"1508.05142","source_kind":"arxiv","source_version":6}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5c456db5c45ed2792ee725e855800b18f3c6b030b2419e7283840630a36ff114","sha256:b32d88a74c572b0b1e3100161dcb07e5841c9fbf698b91765d2a1225143c65bc"],"state_sha256":"af66683d4a2e482dd7a74f663fe7a97b9b6b1b63c67185080750ad965fe087f3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6oIo8bLSpw0CRnk2L1Mjl2L0ya1Sfv3W2EXigXq4lMo5i6xY9GQRWxjxBLO9o9i6dlLV4qJ5LV+8P2FkIf1iCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T08:28:34.865079Z","bundle_sha256":"e34a98adc979c027d1afc7b439575b6ec4185f339810751bfdf87ba3fb36dbe5"}}