{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:EGC54OOOOPZQBRCXFJVFK7ZT42","short_pith_number":"pith:EGC54OOO","canonical_record":{"source":{"id":"1412.8741","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-12-30T19:27:42Z","cross_cats_sorted":[],"title_canon_sha256":"b7daeef634ab79665719b5bac1565add1ad705d16abdfaeb88d8c374d30a2c78","abstract_canon_sha256":"6f0b55a61c55575c26c1740eff4ae43fc5d57ace3455316a7111485d0b89c0e1"},"schema_version":"1.0"},"canonical_sha256":"2185de39ce73f300c4572a6a557f33e6ab0d6d08d8651ed6efdfaf6fd40d9511","source":{"kind":"arxiv","id":"1412.8741","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.8741","created_at":"2026-05-18T02:30:16Z"},{"alias_kind":"arxiv_version","alias_value":"1412.8741v1","created_at":"2026-05-18T02:30:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.8741","created_at":"2026-05-18T02:30:16Z"},{"alias_kind":"pith_short_12","alias_value":"EGC54OOOOPZQ","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_16","alias_value":"EGC54OOOOPZQBRCX","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_8","alias_value":"EGC54OOO","created_at":"2026-05-18T12:28:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:EGC54OOOOPZQBRCXFJVFK7ZT42","target":"record","payload":{"canonical_record":{"source":{"id":"1412.8741","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-12-30T19:27:42Z","cross_cats_sorted":[],"title_canon_sha256":"b7daeef634ab79665719b5bac1565add1ad705d16abdfaeb88d8c374d30a2c78","abstract_canon_sha256":"6f0b55a61c55575c26c1740eff4ae43fc5d57ace3455316a7111485d0b89c0e1"},"schema_version":"1.0"},"canonical_sha256":"2185de39ce73f300c4572a6a557f33e6ab0d6d08d8651ed6efdfaf6fd40d9511","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:30:16.636126Z","signature_b64":"sfYnPLx8z8/hvZ8IeiWosz7yHoSf1VP4qr78VvbWhoAidbswKmIwnVDG2IbIm2Pqygf2PxaFidaI7XJ3wtzvCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2185de39ce73f300c4572a6a557f33e6ab0d6d08d8651ed6efdfaf6fd40d9511","last_reissued_at":"2026-05-18T02:30:16.635693Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:30:16.635693Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1412.8741","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":"2026-05-18T02:30:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EpQbLU1jjCWxYav9FgOgH/5P2eoMJ9G9MdMDi45RpiUWY+G/BveqWTWgGFnb3nKbHanVK0kjlEzPk8wcg87FAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T19:48:47.983221Z"},"content_sha256":"48aa7d1531dd13e094e86e5ca18b4b7ae1fffd558f6514dbebd79f8e2cdd912d","schema_version":"1.0","event_id":"sha256:48aa7d1531dd13e094e86e5ca18b4b7ae1fffd558f6514dbebd79f8e2cdd912d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:EGC54OOOOPZQBRCXFJVFK7ZT42","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A sharper threshold for random groups at density one-half","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Andrew P. S\\'anchez, John M. Mackay, Kasia Jankiewicz, Moon Duchin, Samuel Leli\\`evre, Shelby C. Kilmer","submitted_at":"2014-12-30T19:27:42Z","abstract_excerpt":"In the density model of random groups, we consider presentations with any fixed number m of generators and many random relators of length l, sending l to infinity. If d is a \"density\" parameter measuring the rate of exponential growth of the number of relators compared to the length of relators, then many group-theoretic properties become generically true or generically false at different values of d. The signature theorem for this density model is a phase transition from triviality to hyperbolicity: for d < 1/2, random groups are a.a.s. infinite hyperbolic, while for d > 1/2, random groups ar"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.8741","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":"2026-05-18T02:30:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1lG3CqOxArz0qKc2T/R4UJrgTZFc/sky8oRBBp4zk7ShrUTLlXDLlABoSFbOcTXhSQxudHxeZBZWZDwe6nJRDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T19:48:47.983841Z"},"content_sha256":"524645d1c93c30455bc09e5e6c5a55a9bf1318625bc2fa624576abaa6a7cc929","schema_version":"1.0","event_id":"sha256:524645d1c93c30455bc09e5e6c5a55a9bf1318625bc2fa624576abaa6a7cc929"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EGC54OOOOPZQBRCXFJVFK7ZT42/bundle.json","state_url":"https://pith.science/pith/EGC54OOOOPZQBRCXFJVFK7ZT42/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EGC54OOOOPZQBRCXFJVFK7ZT42/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":"2026-05-27T19:48:47Z","links":{"resolver":"https://pith.science/pith/EGC54OOOOPZQBRCXFJVFK7ZT42","bundle":"https://pith.science/pith/EGC54OOOOPZQBRCXFJVFK7ZT42/bundle.json","state":"https://pith.science/pith/EGC54OOOOPZQBRCXFJVFK7ZT42/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EGC54OOOOPZQBRCXFJVFK7ZT42/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:EGC54OOOOPZQBRCXFJVFK7ZT42","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":"6f0b55a61c55575c26c1740eff4ae43fc5d57ace3455316a7111485d0b89c0e1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-12-30T19:27:42Z","title_canon_sha256":"b7daeef634ab79665719b5bac1565add1ad705d16abdfaeb88d8c374d30a2c78"},"schema_version":"1.0","source":{"id":"1412.8741","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.8741","created_at":"2026-05-18T02:30:16Z"},{"alias_kind":"arxiv_version","alias_value":"1412.8741v1","created_at":"2026-05-18T02:30:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.8741","created_at":"2026-05-18T02:30:16Z"},{"alias_kind":"pith_short_12","alias_value":"EGC54OOOOPZQ","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_16","alias_value":"EGC54OOOOPZQBRCX","created_at":"2026-05-18T12:28:25Z"},{"alias_kind":"pith_short_8","alias_value":"EGC54OOO","created_at":"2026-05-18T12:28:25Z"}],"graph_snapshots":[{"event_id":"sha256:524645d1c93c30455bc09e5e6c5a55a9bf1318625bc2fa624576abaa6a7cc929","target":"graph","created_at":"2026-05-18T02:30:16Z","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":"In the density model of random groups, we consider presentations with any fixed number m of generators and many random relators of length l, sending l to infinity. If d is a \"density\" parameter measuring the rate of exponential growth of the number of relators compared to the length of relators, then many group-theoretic properties become generically true or generically false at different values of d. The signature theorem for this density model is a phase transition from triviality to hyperbolicity: for d < 1/2, random groups are a.a.s. infinite hyperbolic, while for d > 1/2, random groups ar","authors_text":"Andrew P. S\\'anchez, John M. Mackay, Kasia Jankiewicz, Moon Duchin, Samuel Leli\\`evre, Shelby C. Kilmer","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-12-30T19:27:42Z","title":"A sharper threshold for random groups at density one-half"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.8741","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":"sha256:48aa7d1531dd13e094e86e5ca18b4b7ae1fffd558f6514dbebd79f8e2cdd912d","target":"record","created_at":"2026-05-18T02:30:16Z","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":"6f0b55a61c55575c26c1740eff4ae43fc5d57ace3455316a7111485d0b89c0e1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-12-30T19:27:42Z","title_canon_sha256":"b7daeef634ab79665719b5bac1565add1ad705d16abdfaeb88d8c374d30a2c78"},"schema_version":"1.0","source":{"id":"1412.8741","kind":"arxiv","version":1}},"canonical_sha256":"2185de39ce73f300c4572a6a557f33e6ab0d6d08d8651ed6efdfaf6fd40d9511","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2185de39ce73f300c4572a6a557f33e6ab0d6d08d8651ed6efdfaf6fd40d9511","first_computed_at":"2026-05-18T02:30:16.635693Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:30:16.635693Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sfYnPLx8z8/hvZ8IeiWosz7yHoSf1VP4qr78VvbWhoAidbswKmIwnVDG2IbIm2Pqygf2PxaFidaI7XJ3wtzvCw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:30:16.636126Z","signed_message":"canonical_sha256_bytes"},"source_id":"1412.8741","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:48aa7d1531dd13e094e86e5ca18b4b7ae1fffd558f6514dbebd79f8e2cdd912d","sha256:524645d1c93c30455bc09e5e6c5a55a9bf1318625bc2fa624576abaa6a7cc929"],"state_sha256":"d4d06aee8a53dac28188d06814b63cd3e4de62bc90455a56d55163afd8e3cf95"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lQFlnJscC8BuptT6N9XIpkTxyYzjoZbaTbS3XBD2vjzwA91cmrIEvobV/6a8u8xGUtUl/0ClSIPgyrsmH6dFCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T19:48:47.986619Z","bundle_sha256":"142e9da06577bc496fed28caab4dd1ef0dd25a01dcd955d397c74f52ce45d3f6"}}