{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:1996:ZKYWUQYW2Z2CL5GQSEJ4FF7LVR","short_pith_number":"pith:ZKYWUQYW","canonical_record":{"source":{"id":"math/9612211","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.GR","submitted_at":"1996-12-03T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"b25b1943a0e2b9e458dd8548f86f27c4431dc8d36df8a94097fc06542bbf6344","abstract_canon_sha256":"ee130ff158920aa229e60e737b18390c053aefc10adc895cfbac2539760bb945"},"schema_version":"1.0"},"canonical_sha256":"cab16a4316d67425f4d09113c297ebac651b79d92c4cdd42835f6f141ec6a153","source":{"kind":"arxiv","id":"math/9612211","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9612211","created_at":"2026-05-18T01:05:37Z"},{"alias_kind":"arxiv_version","alias_value":"math/9612211v1","created_at":"2026-05-18T01:05:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9612211","created_at":"2026-05-18T01:05:37Z"},{"alias_kind":"pith_short_12","alias_value":"ZKYWUQYW2Z2C","created_at":"2026-05-18T12:25:48Z"},{"alias_kind":"pith_short_16","alias_value":"ZKYWUQYW2Z2CL5GQ","created_at":"2026-05-18T12:25:48Z"},{"alias_kind":"pith_short_8","alias_value":"ZKYWUQYW","created_at":"2026-05-18T12:25:48Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:1996:ZKYWUQYW2Z2CL5GQSEJ4FF7LVR","target":"record","payload":{"canonical_record":{"source":{"id":"math/9612211","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.GR","submitted_at":"1996-12-03T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"b25b1943a0e2b9e458dd8548f86f27c4431dc8d36df8a94097fc06542bbf6344","abstract_canon_sha256":"ee130ff158920aa229e60e737b18390c053aefc10adc895cfbac2539760bb945"},"schema_version":"1.0"},"canonical_sha256":"cab16a4316d67425f4d09113c297ebac651b79d92c4cdd42835f6f141ec6a153","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:37.390979Z","signature_b64":"8FNtt1l917HU6OOo2xhV/afJgLbRmzceaeQD8jPeEL9RnAnfpkAag0c5zxGn/YLdbaYlXbksSWVxJYDGiU3kAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cab16a4316d67425f4d09113c297ebac651b79d92c4cdd42835f6f141ec6a153","last_reissued_at":"2026-05-18T01:05:37.390468Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:37.390468Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/9612211","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":"2026-05-18T01:05:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fAQ1DmoGjcBirFM1+BG91TDOfdIsgeyMXVTDwhPobIZcrejD51EE0YaAlyw8t/OZmI1ryqHhUd5mkQIDMkq6Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T03:55:41.045958Z"},"content_sha256":"59a7cd32b46766887fec3456110bbeaf30cc2412ab14800f04d2574f8699cd1e","schema_version":"1.0","event_id":"sha256:59a7cd32b46766887fec3456110bbeaf30cc2412ab14800f04d2574f8699cd1e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:1996:ZKYWUQYW2Z2CL5GQSEJ4FF7LVR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Ping-Pong on Negatively Curved Groups","license":"","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Rita Gitik","submitted_at":"1996-12-03T00:00:00Z","abstract_excerpt":"We prove several generalisations of the ping-pong lemma for negatively curved groups."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9612211","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":"2026-05-18T01:05:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tQXIlxxLaMkl5zzHHhqoAjxqwlqIgXTIfgFU2nC/Y6wHW6reYlsmFaftGPAJDNID3S6oBBhydlWiKJdvAzOQCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T03:55:41.046327Z"},"content_sha256":"14ebef9bf89571aca34fad1476e48525223c05a80527e300a65c64474b8cc4bc","schema_version":"1.0","event_id":"sha256:14ebef9bf89571aca34fad1476e48525223c05a80527e300a65c64474b8cc4bc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZKYWUQYW2Z2CL5GQSEJ4FF7LVR/bundle.json","state_url":"https://pith.science/pith/ZKYWUQYW2Z2CL5GQSEJ4FF7LVR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZKYWUQYW2Z2CL5GQSEJ4FF7LVR/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":"2026-06-22T03:55:41Z","links":{"resolver":"https://pith.science/pith/ZKYWUQYW2Z2CL5GQSEJ4FF7LVR","bundle":"https://pith.science/pith/ZKYWUQYW2Z2CL5GQSEJ4FF7LVR/bundle.json","state":"https://pith.science/pith/ZKYWUQYW2Z2CL5GQSEJ4FF7LVR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZKYWUQYW2Z2CL5GQSEJ4FF7LVR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1996:ZKYWUQYW2Z2CL5GQSEJ4FF7LVR","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":"ee130ff158920aa229e60e737b18390c053aefc10adc895cfbac2539760bb945","cross_cats_sorted":[],"license":"","primary_cat":"math.GR","submitted_at":"1996-12-03T00:00:00Z","title_canon_sha256":"b25b1943a0e2b9e458dd8548f86f27c4431dc8d36df8a94097fc06542bbf6344"},"schema_version":"1.0","source":{"id":"math/9612211","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9612211","created_at":"2026-05-18T01:05:37Z"},{"alias_kind":"arxiv_version","alias_value":"math/9612211v1","created_at":"2026-05-18T01:05:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9612211","created_at":"2026-05-18T01:05:37Z"},{"alias_kind":"pith_short_12","alias_value":"ZKYWUQYW2Z2C","created_at":"2026-05-18T12:25:48Z"},{"alias_kind":"pith_short_16","alias_value":"ZKYWUQYW2Z2CL5GQ","created_at":"2026-05-18T12:25:48Z"},{"alias_kind":"pith_short_8","alias_value":"ZKYWUQYW","created_at":"2026-05-18T12:25:48Z"}],"graph_snapshots":[{"event_id":"sha256:14ebef9bf89571aca34fad1476e48525223c05a80527e300a65c64474b8cc4bc","target":"graph","created_at":"2026-05-18T01:05:37Z","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 several generalisations of the ping-pong lemma for negatively curved groups.","authors_text":"Rita Gitik","cross_cats":[],"headline":"","license":"","primary_cat":"math.GR","submitted_at":"1996-12-03T00:00:00Z","title":"Ping-Pong on Negatively Curved Groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9612211","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":"sha256:59a7cd32b46766887fec3456110bbeaf30cc2412ab14800f04d2574f8699cd1e","target":"record","created_at":"2026-05-18T01:05:37Z","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":"ee130ff158920aa229e60e737b18390c053aefc10adc895cfbac2539760bb945","cross_cats_sorted":[],"license":"","primary_cat":"math.GR","submitted_at":"1996-12-03T00:00:00Z","title_canon_sha256":"b25b1943a0e2b9e458dd8548f86f27c4431dc8d36df8a94097fc06542bbf6344"},"schema_version":"1.0","source":{"id":"math/9612211","kind":"arxiv","version":1}},"canonical_sha256":"cab16a4316d67425f4d09113c297ebac651b79d92c4cdd42835f6f141ec6a153","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cab16a4316d67425f4d09113c297ebac651b79d92c4cdd42835f6f141ec6a153","first_computed_at":"2026-05-18T01:05:37.390468Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:37.390468Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8FNtt1l917HU6OOo2xhV/afJgLbRmzceaeQD8jPeEL9RnAnfpkAag0c5zxGn/YLdbaYlXbksSWVxJYDGiU3kAw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:37.390979Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9612211","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:59a7cd32b46766887fec3456110bbeaf30cc2412ab14800f04d2574f8699cd1e","sha256:14ebef9bf89571aca34fad1476e48525223c05a80527e300a65c64474b8cc4bc"],"state_sha256":"c39bbb96f2cf9b1ff4acd6c55d7497b8e1ed18208ee6e83280ad57afce5babc2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JuwEgwgpdb3ZZcvzGr/HefSFCC+8kEcOy9YVTF7Z+aPk6eKyBAz5nspwwY0zHmM3UoDZb3ZZyCsXS/QAENpNDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T03:55:41.048607Z","bundle_sha256":"833d66c49c5a395db55b42f891e2118d86ca5a09da09c9157f0b0d1eceeaa56f"}}