{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:ZW27HYZ6PSM33DWDARJXXX4OHO","short_pith_number":"pith:ZW27HYZ6","canonical_record":{"source":{"id":"1601.07520","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-01-27T19:54:36Z","cross_cats_sorted":["math.CO","math.GR","math.PR"],"title_canon_sha256":"a8acd631e3a8f0bfa921eec7d38dc21168f6342611f1935ce82163c7d1144696","abstract_canon_sha256":"805fc4ff6a42ebe775342ef3e15a08ada786071808c2e61072539accc0290737"},"schema_version":"1.0"},"canonical_sha256":"cdb5f3e33e7c99bd8ec304537bdf8e3b8ca00b500f16a9f75bd32377f445cf2f","source":{"kind":"arxiv","id":"1601.07520","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.07520","created_at":"2026-05-18T00:33:07Z"},{"alias_kind":"arxiv_version","alias_value":"1601.07520v3","created_at":"2026-05-18T00:33:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.07520","created_at":"2026-05-18T00:33:07Z"},{"alias_kind":"pith_short_12","alias_value":"ZW27HYZ6PSM3","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_16","alias_value":"ZW27HYZ6PSM33DWD","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_8","alias_value":"ZW27HYZ6","created_at":"2026-05-18T12:30:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:ZW27HYZ6PSM33DWDARJXXX4OHO","target":"record","payload":{"canonical_record":{"source":{"id":"1601.07520","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-01-27T19:54:36Z","cross_cats_sorted":["math.CO","math.GR","math.PR"],"title_canon_sha256":"a8acd631e3a8f0bfa921eec7d38dc21168f6342611f1935ce82163c7d1144696","abstract_canon_sha256":"805fc4ff6a42ebe775342ef3e15a08ada786071808c2e61072539accc0290737"},"schema_version":"1.0"},"canonical_sha256":"cdb5f3e33e7c99bd8ec304537bdf8e3b8ca00b500f16a9f75bd32377f445cf2f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:33:07.320012Z","signature_b64":"cu4DZHFDBeuRUeI9GLg+5yerUGls/SaodSZqPxCBuVziTrzRgyNYQuLxBLgUul5pKQjXXIF0+oCrTMFo45jECQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cdb5f3e33e7c99bd8ec304537bdf8e3b8ca00b500f16a9f75bd32377f445cf2f","last_reissued_at":"2026-05-18T00:33:07.319395Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:33:07.319395Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1601.07520","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":"2026-05-18T00:33:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"C/y0kgjrjkqwcMNAKG3ivQkks2HXMMsx0QUyUGsspvC/ZMZ6ZfKTT9nhj1cBKN6tbwwcY88UPNGbprykzMUOCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T08:28:35.442309Z"},"content_sha256":"7e9d056041ab2888d88740c9958a63ad533ec75a6f558a05ac49d19dbe8148b2","schema_version":"1.0","event_id":"sha256:7e9d056041ab2888d88740c9958a63ad533ec75a6f558a05ac49d19dbe8148b2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:ZW27HYZ6PSM33DWDARJXXX4OHO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On freeness of the random fundamental group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.GR","math.PR"],"primary_cat":"math.AT","authors_text":"Andrew Newman","submitted_at":"2016-01-27T19:54:36Z","abstract_excerpt":"Let $Y(n, p)$ denote the probability space of random 2-dimensional simplicial complexes in the Linial--Meshulam model, and let $Y \\sim Y(n, p)$ denote a random complex chosen according to this distribution. In a paper of Cohen, Costa, Farber, and Kappeler, it is shown that for $p = o(1/n)$ with high probability $\\pi_1(Y)$ is free. Following that, a paper of Costa and Farber shows that for values of $p$ which satisfy $3/n < p \\ll n^{-46/47}$, with high probability $\\pi_1(Y)$ is not free. Here we improve on both of these results to show that there are explicit constants $\\gamma_2 < c_2 < 3$, so "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.07520","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":"2026-05-18T00:33:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OzfP00ryQgnaZwvJVHLJRtiBql+Klm3X7PkjO1miAzxz1PGpsb72Z9nzaxPmwmJNrIsAztyc35aOVjuMShV/DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T08:28:35.442687Z"},"content_sha256":"ca8f864ca25f9748655aeefb3d63ceae5c201cacdbad475f89a03a183b6a13e5","schema_version":"1.0","event_id":"sha256:ca8f864ca25f9748655aeefb3d63ceae5c201cacdbad475f89a03a183b6a13e5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZW27HYZ6PSM33DWDARJXXX4OHO/bundle.json","state_url":"https://pith.science/pith/ZW27HYZ6PSM33DWDARJXXX4OHO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZW27HYZ6PSM33DWDARJXXX4OHO/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":"2026-06-02T08:28:35Z","links":{"resolver":"https://pith.science/pith/ZW27HYZ6PSM33DWDARJXXX4OHO","bundle":"https://pith.science/pith/ZW27HYZ6PSM33DWDARJXXX4OHO/bundle.json","state":"https://pith.science/pith/ZW27HYZ6PSM33DWDARJXXX4OHO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZW27HYZ6PSM33DWDARJXXX4OHO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:ZW27HYZ6PSM33DWDARJXXX4OHO","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":"805fc4ff6a42ebe775342ef3e15a08ada786071808c2e61072539accc0290737","cross_cats_sorted":["math.CO","math.GR","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-01-27T19:54:36Z","title_canon_sha256":"a8acd631e3a8f0bfa921eec7d38dc21168f6342611f1935ce82163c7d1144696"},"schema_version":"1.0","source":{"id":"1601.07520","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.07520","created_at":"2026-05-18T00:33:07Z"},{"alias_kind":"arxiv_version","alias_value":"1601.07520v3","created_at":"2026-05-18T00:33:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.07520","created_at":"2026-05-18T00:33:07Z"},{"alias_kind":"pith_short_12","alias_value":"ZW27HYZ6PSM3","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_16","alias_value":"ZW27HYZ6PSM33DWD","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_8","alias_value":"ZW27HYZ6","created_at":"2026-05-18T12:30:55Z"}],"graph_snapshots":[{"event_id":"sha256:ca8f864ca25f9748655aeefb3d63ceae5c201cacdbad475f89a03a183b6a13e5","target":"graph","created_at":"2026-05-18T00:33:07Z","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 $Y(n, p)$ denote the probability space of random 2-dimensional simplicial complexes in the Linial--Meshulam model, and let $Y \\sim Y(n, p)$ denote a random complex chosen according to this distribution. In a paper of Cohen, Costa, Farber, and Kappeler, it is shown that for $p = o(1/n)$ with high probability $\\pi_1(Y)$ is free. Following that, a paper of Costa and Farber shows that for values of $p$ which satisfy $3/n < p \\ll n^{-46/47}$, with high probability $\\pi_1(Y)$ is not free. Here we improve on both of these results to show that there are explicit constants $\\gamma_2 < c_2 < 3$, so ","authors_text":"Andrew Newman","cross_cats":["math.CO","math.GR","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-01-27T19:54:36Z","title":"On freeness of the random fundamental group"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.07520","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":"sha256:7e9d056041ab2888d88740c9958a63ad533ec75a6f558a05ac49d19dbe8148b2","target":"record","created_at":"2026-05-18T00:33:07Z","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":"805fc4ff6a42ebe775342ef3e15a08ada786071808c2e61072539accc0290737","cross_cats_sorted":["math.CO","math.GR","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-01-27T19:54:36Z","title_canon_sha256":"a8acd631e3a8f0bfa921eec7d38dc21168f6342611f1935ce82163c7d1144696"},"schema_version":"1.0","source":{"id":"1601.07520","kind":"arxiv","version":3}},"canonical_sha256":"cdb5f3e33e7c99bd8ec304537bdf8e3b8ca00b500f16a9f75bd32377f445cf2f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cdb5f3e33e7c99bd8ec304537bdf8e3b8ca00b500f16a9f75bd32377f445cf2f","first_computed_at":"2026-05-18T00:33:07.319395Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:33:07.319395Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cu4DZHFDBeuRUeI9GLg+5yerUGls/SaodSZqPxCBuVziTrzRgyNYQuLxBLgUul5pKQjXXIF0+oCrTMFo45jECQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:33:07.320012Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.07520","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7e9d056041ab2888d88740c9958a63ad533ec75a6f558a05ac49d19dbe8148b2","sha256:ca8f864ca25f9748655aeefb3d63ceae5c201cacdbad475f89a03a183b6a13e5"],"state_sha256":"2c435b88ecbf05af1da8c26ad264e2b25212dc3d60dd972c97dea468ca43bb36"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3wxGLYNbwERLIU9OupkJZtcXXSXVJ5dpWvVsCE4UGCDs2oBIG6mUYXOfsUkZptQnxRfYwUIPgJg2xgsFHrpvDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T08:28:35.444612Z","bundle_sha256":"5cb263acdf7e903182b5bf78aa78ad54b13df41fe475ee7582690e3448f30462"}}