{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:W3IYHORI4UEDAMHT4ZTOXAO34A","short_pith_number":"pith:W3IYHORI","canonical_record":{"source":{"id":"1501.06682","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-01-27T08:11:17Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"00f7bb680bbe517ad1674c3b7490550059c5946c1a955b04605bd3d524193c48","abstract_canon_sha256":"736fcb48f68c41f6250de74878ad92a87f7e75dd8dc35b3fec0d688d72ce83a7"},"schema_version":"1.0"},"canonical_sha256":"b6d183ba28e5083030f3e666eb81dbe0251f7ae5f2674b0baff8e9c2904edabc","source":{"kind":"arxiv","id":"1501.06682","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.06682","created_at":"2026-05-18T02:17:42Z"},{"alias_kind":"arxiv_version","alias_value":"1501.06682v2","created_at":"2026-05-18T02:17:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.06682","created_at":"2026-05-18T02:17:42Z"},{"alias_kind":"pith_short_12","alias_value":"W3IYHORI4UED","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"W3IYHORI4UEDAMHT","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"W3IYHORI","created_at":"2026-05-18T12:29:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:W3IYHORI4UEDAMHT4ZTOXAO34A","target":"record","payload":{"canonical_record":{"source":{"id":"1501.06682","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-01-27T08:11:17Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"00f7bb680bbe517ad1674c3b7490550059c5946c1a955b04605bd3d524193c48","abstract_canon_sha256":"736fcb48f68c41f6250de74878ad92a87f7e75dd8dc35b3fec0d688d72ce83a7"},"schema_version":"1.0"},"canonical_sha256":"b6d183ba28e5083030f3e666eb81dbe0251f7ae5f2674b0baff8e9c2904edabc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:17:42.534529Z","signature_b64":"piEub0zDqZtoCcpYATpc9UG+ghcfoHIcYUaqvRcW3URbOaNG2HXtrFmZLVsAqEbePv/0O4rhjj+uRKSEJlxFDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b6d183ba28e5083030f3e666eb81dbe0251f7ae5f2674b0baff8e9c2904edabc","last_reissued_at":"2026-05-18T02:17:42.534149Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:17:42.534149Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1501.06682","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:17:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"usXEtzbNXtwe6SEZghKCGWTMRuv4rS5elv9u0w1UXanqw82INpCTTvmP1lmBMsDPni/o1It5G0PbH7NFu342DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T00:26:17.451983Z"},"content_sha256":"633f47822b8d955b748201b39b78b4207124c04405c7e36c06421b6c4eaeff25","schema_version":"1.0","event_id":"sha256:633f47822b8d955b748201b39b78b4207124c04405c7e36c06421b6c4eaeff25"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:W3IYHORI4UEDAMHT4ZTOXAO34A","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The $\\Sigma$-invariants of Thompson's group $F$ via Morse theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GR","authors_text":"Matthew C. B. Zaremsky, Stefan Witzel","submitted_at":"2015-01-27T08:11:17Z","abstract_excerpt":"Bieri, Geoghegan and Kochloukova computed the BNSR-invariants $\\Sigma^m(F)$ of Thompson's group $F$ for all $m$. We recompute these using entirely geometric techniques, making use of the Stein--Farley CAT(0) cube complex on which $F$ acts."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.06682","kind":"arxiv","version":2},"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":"2026-05-18T02:17:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XjS0TQcZQFIuKRgT9U+4dgdIPoIty2X8DsoCa3uUnZZi4QCC+uexoacGwZy4z4KNG0kn3Sr1W6IpLSNXEN14Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T00:26:17.452347Z"},"content_sha256":"6f033a927cce0a921468707607619445a347a4c1f4cb5f5a13725afa68f3f57c","schema_version":"1.0","event_id":"sha256:6f033a927cce0a921468707607619445a347a4c1f4cb5f5a13725afa68f3f57c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/W3IYHORI4UEDAMHT4ZTOXAO34A/bundle.json","state_url":"https://pith.science/pith/W3IYHORI4UEDAMHT4ZTOXAO34A/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/W3IYHORI4UEDAMHT4ZTOXAO34A/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":"2026-05-31T00:26:17Z","links":{"resolver":"https://pith.science/pith/W3IYHORI4UEDAMHT4ZTOXAO34A","bundle":"https://pith.science/pith/W3IYHORI4UEDAMHT4ZTOXAO34A/bundle.json","state":"https://pith.science/pith/W3IYHORI4UEDAMHT4ZTOXAO34A/state.json","well_known_bundle":"https://pith.science/.well-known/pith/W3IYHORI4UEDAMHT4ZTOXAO34A/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:W3IYHORI4UEDAMHT4ZTOXAO34A","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":"736fcb48f68c41f6250de74878ad92a87f7e75dd8dc35b3fec0d688d72ce83a7","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-01-27T08:11:17Z","title_canon_sha256":"00f7bb680bbe517ad1674c3b7490550059c5946c1a955b04605bd3d524193c48"},"schema_version":"1.0","source":{"id":"1501.06682","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.06682","created_at":"2026-05-18T02:17:42Z"},{"alias_kind":"arxiv_version","alias_value":"1501.06682v2","created_at":"2026-05-18T02:17:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.06682","created_at":"2026-05-18T02:17:42Z"},{"alias_kind":"pith_short_12","alias_value":"W3IYHORI4UED","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"W3IYHORI4UEDAMHT","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"W3IYHORI","created_at":"2026-05-18T12:29:47Z"}],"graph_snapshots":[{"event_id":"sha256:6f033a927cce0a921468707607619445a347a4c1f4cb5f5a13725afa68f3f57c","target":"graph","created_at":"2026-05-18T02:17:42Z","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":"Bieri, Geoghegan and Kochloukova computed the BNSR-invariants $\\Sigma^m(F)$ of Thompson's group $F$ for all $m$. We recompute these using entirely geometric techniques, making use of the Stein--Farley CAT(0) cube complex on which $F$ acts.","authors_text":"Matthew C. B. Zaremsky, Stefan Witzel","cross_cats":["math.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-01-27T08:11:17Z","title":"The $\\Sigma$-invariants of Thompson's group $F$ via Morse theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.06682","kind":"arxiv","version":2},"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":"sha256:633f47822b8d955b748201b39b78b4207124c04405c7e36c06421b6c4eaeff25","target":"record","created_at":"2026-05-18T02:17:42Z","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":"736fcb48f68c41f6250de74878ad92a87f7e75dd8dc35b3fec0d688d72ce83a7","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-01-27T08:11:17Z","title_canon_sha256":"00f7bb680bbe517ad1674c3b7490550059c5946c1a955b04605bd3d524193c48"},"schema_version":"1.0","source":{"id":"1501.06682","kind":"arxiv","version":2}},"canonical_sha256":"b6d183ba28e5083030f3e666eb81dbe0251f7ae5f2674b0baff8e9c2904edabc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b6d183ba28e5083030f3e666eb81dbe0251f7ae5f2674b0baff8e9c2904edabc","first_computed_at":"2026-05-18T02:17:42.534149Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:17:42.534149Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"piEub0zDqZtoCcpYATpc9UG+ghcfoHIcYUaqvRcW3URbOaNG2HXtrFmZLVsAqEbePv/0O4rhjj+uRKSEJlxFDA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:17:42.534529Z","signed_message":"canonical_sha256_bytes"},"source_id":"1501.06682","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:633f47822b8d955b748201b39b78b4207124c04405c7e36c06421b6c4eaeff25","sha256:6f033a927cce0a921468707607619445a347a4c1f4cb5f5a13725afa68f3f57c"],"state_sha256":"6db1a0d9cf59ad4021185b6d9cbdf5d43d5e07c7e5a021b7a1913c38572ca18c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CnLIrfRqEZfeREtn7V2xlZc61Y1Aw6fyxnHfW6XJk4G8fZRpRTqYNvgwyOcIUcQHb/nzX90zKzV6ape4K2IODg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T00:26:17.454506Z","bundle_sha256":"7eeca431ce8f9bee983609de68fcad7dd7e048482b9ce07d26e1149b418b0243"}}