{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:ABWW7GKTZ2SMUFKIIRYOXVTYZO","short_pith_number":"pith:ABWW7GKT","canonical_record":{"source":{"id":"1507.08597","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-07-30T17:51:24Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"e5a8ca22e906ee877011ee6581e4b42da2f6b6d08cf83c57edf10bcfb2255a53","abstract_canon_sha256":"0029c4001e26bd5b0de80378df98623e18f723846c77fc466e93fe00ae37cf76"},"schema_version":"1.0"},"canonical_sha256":"006d6f9953cea4ca15484470ebd678cbb900488e90b28631054221cdea4fc388","source":{"kind":"arxiv","id":"1507.08597","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.08597","created_at":"2026-05-18T01:36:06Z"},{"alias_kind":"arxiv_version","alias_value":"1507.08597v1","created_at":"2026-05-18T01:36:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.08597","created_at":"2026-05-18T01:36:06Z"},{"alias_kind":"pith_short_12","alias_value":"ABWW7GKTZ2SM","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_16","alias_value":"ABWW7GKTZ2SMUFKI","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_8","alias_value":"ABWW7GKT","created_at":"2026-05-18T12:29:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:ABWW7GKTZ2SMUFKIIRYOXVTYZO","target":"record","payload":{"canonical_record":{"source":{"id":"1507.08597","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-07-30T17:51:24Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"e5a8ca22e906ee877011ee6581e4b42da2f6b6d08cf83c57edf10bcfb2255a53","abstract_canon_sha256":"0029c4001e26bd5b0de80378df98623e18f723846c77fc466e93fe00ae37cf76"},"schema_version":"1.0"},"canonical_sha256":"006d6f9953cea4ca15484470ebd678cbb900488e90b28631054221cdea4fc388","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:06.032092Z","signature_b64":"pWC25PNsGH+aEEQB9g+x1swRPimp9qqdcU5+GNW3+cAPRiwpwojUtPmJMzeleG83UGiE5lqe5SeriOYAJki8Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"006d6f9953cea4ca15484470ebd678cbb900488e90b28631054221cdea4fc388","last_reissued_at":"2026-05-18T01:36:06.031651Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:06.031651Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.08597","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":"2026-05-18T01:36:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Vx+WXXH41cXbXRqslQmNdyYkWkj06kjqp9rqP7re1YQlKAukqtZlrxMTu3kFx/K0F8qyiY0CHiZpjWTrQaO7BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T14:06:03.758366Z"},"content_sha256":"e9fd5698f29e0562e277497f7e8783d95e6a66627529255a9f614b340af2c2b2","schema_version":"1.0","event_id":"sha256:e9fd5698f29e0562e277497f7e8783d95e6a66627529255a9f614b340af2c2b2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:ABWW7GKTZ2SMUFKIIRYOXVTYZO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Separation in the BNSR-invariants of the pure braid groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GR","authors_text":"Matthew C. B. Zaremsky","submitted_at":"2015-07-30T17:51:24Z","abstract_excerpt":"We inspect the BNSR-invariants $\\Sigma^m(P_n)$ of the pure braid groups $P_n$, using Morse theory. The BNS-invariants $\\Sigma^1(P_n)$ were previously computed by Koban, McCammond and Meier. We prove that for any $3\\le m\\le n$, the inclusion $\\Sigma^{m-2}(P_n)\\subseteq \\Sigma^{m-3}(P_n)$ is proper, but $\\Sigma^\\infty(P_n)=\\Sigma^{n-2}(P_n)$. We write down explicit character classes in each relevant $\\Sigma^{m-3}(P_n)\\setminus \\Sigma^{m-2}(P_n)$. In particular we get examples of normal subgroups $N\\le P_n$ with $P_n/N\\cong\\mathbb{Z}$ such that $N$ is of type $F_{m-3}$ but not $F_{m-2}$, for all "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08597","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":"2026-05-18T01:36:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Dl62/trgnvPdoH8AYvMu+gMKWx9yikAd9StF6JpfJr2MyKODFNrqGAnlb9GVpDg3EiF4DoScUGvuMHIRfPIDAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T14:06:03.759054Z"},"content_sha256":"169f3d96b44d21bf22038274ca0875e2bee6ce9dc492d0a82ffb69d35bc5cff3","schema_version":"1.0","event_id":"sha256:169f3d96b44d21bf22038274ca0875e2bee6ce9dc492d0a82ffb69d35bc5cff3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ABWW7GKTZ2SMUFKIIRYOXVTYZO/bundle.json","state_url":"https://pith.science/pith/ABWW7GKTZ2SMUFKIIRYOXVTYZO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ABWW7GKTZ2SMUFKIIRYOXVTYZO/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":"2026-06-07T14:06:03Z","links":{"resolver":"https://pith.science/pith/ABWW7GKTZ2SMUFKIIRYOXVTYZO","bundle":"https://pith.science/pith/ABWW7GKTZ2SMUFKIIRYOXVTYZO/bundle.json","state":"https://pith.science/pith/ABWW7GKTZ2SMUFKIIRYOXVTYZO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ABWW7GKTZ2SMUFKIIRYOXVTYZO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:ABWW7GKTZ2SMUFKIIRYOXVTYZO","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":"0029c4001e26bd5b0de80378df98623e18f723846c77fc466e93fe00ae37cf76","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-07-30T17:51:24Z","title_canon_sha256":"e5a8ca22e906ee877011ee6581e4b42da2f6b6d08cf83c57edf10bcfb2255a53"},"schema_version":"1.0","source":{"id":"1507.08597","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.08597","created_at":"2026-05-18T01:36:06Z"},{"alias_kind":"arxiv_version","alias_value":"1507.08597v1","created_at":"2026-05-18T01:36:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.08597","created_at":"2026-05-18T01:36:06Z"},{"alias_kind":"pith_short_12","alias_value":"ABWW7GKTZ2SM","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_16","alias_value":"ABWW7GKTZ2SMUFKI","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_8","alias_value":"ABWW7GKT","created_at":"2026-05-18T12:29:10Z"}],"graph_snapshots":[{"event_id":"sha256:169f3d96b44d21bf22038274ca0875e2bee6ce9dc492d0a82ffb69d35bc5cff3","target":"graph","created_at":"2026-05-18T01:36:06Z","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 inspect the BNSR-invariants $\\Sigma^m(P_n)$ of the pure braid groups $P_n$, using Morse theory. The BNS-invariants $\\Sigma^1(P_n)$ were previously computed by Koban, McCammond and Meier. We prove that for any $3\\le m\\le n$, the inclusion $\\Sigma^{m-2}(P_n)\\subseteq \\Sigma^{m-3}(P_n)$ is proper, but $\\Sigma^\\infty(P_n)=\\Sigma^{n-2}(P_n)$. We write down explicit character classes in each relevant $\\Sigma^{m-3}(P_n)\\setminus \\Sigma^{m-2}(P_n)$. In particular we get examples of normal subgroups $N\\le P_n$ with $P_n/N\\cong\\mathbb{Z}$ such that $N$ is of type $F_{m-3}$ but not $F_{m-2}$, for all ","authors_text":"Matthew C. B. Zaremsky","cross_cats":["math.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-07-30T17:51:24Z","title":"Separation in the BNSR-invariants of the pure braid groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08597","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:e9fd5698f29e0562e277497f7e8783d95e6a66627529255a9f614b340af2c2b2","target":"record","created_at":"2026-05-18T01:36:06Z","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":"0029c4001e26bd5b0de80378df98623e18f723846c77fc466e93fe00ae37cf76","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-07-30T17:51:24Z","title_canon_sha256":"e5a8ca22e906ee877011ee6581e4b42da2f6b6d08cf83c57edf10bcfb2255a53"},"schema_version":"1.0","source":{"id":"1507.08597","kind":"arxiv","version":1}},"canonical_sha256":"006d6f9953cea4ca15484470ebd678cbb900488e90b28631054221cdea4fc388","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"006d6f9953cea4ca15484470ebd678cbb900488e90b28631054221cdea4fc388","first_computed_at":"2026-05-18T01:36:06.031651Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:36:06.031651Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pWC25PNsGH+aEEQB9g+x1swRPimp9qqdcU5+GNW3+cAPRiwpwojUtPmJMzeleG83UGiE5lqe5SeriOYAJki8Cw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:36:06.032092Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.08597","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e9fd5698f29e0562e277497f7e8783d95e6a66627529255a9f614b340af2c2b2","sha256:169f3d96b44d21bf22038274ca0875e2bee6ce9dc492d0a82ffb69d35bc5cff3"],"state_sha256":"7a4871e5b068260c5947ff98727409a37ad848d1fee3f275f539c6860c84fff7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"j+hhXr9xsCfTlL1Io63ejkb8C3ppLNrTcGG7bQftvj5561ap2lfW5OWK/ejqT6jyJ6WsIvfEq9ci984auRzpDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T14:06:03.763854Z","bundle_sha256":"db4f8c92b971654ab3d5ad6691a75b5f38a70a98b4364de20dfa5f6276a52a0f"}}