{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:3I6QPO673K7HX45ILXSX4UVZRM","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":"99e01f7861059d6707de2d1b40b5a46fedc861edbeab9ee0ebb931cda92cbf1f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2009-07-27T14:34:01Z","title_canon_sha256":"90f706a099ece2c5d448c4fb80943ef4bfd05304e6df3dceff11454728f2b5bb"},"schema_version":"1.0","source":{"id":"0907.4642","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0907.4642","created_at":"2026-05-18T02:57:07Z"},{"alias_kind":"arxiv_version","alias_value":"0907.4642v4","created_at":"2026-05-18T02:57:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0907.4642","created_at":"2026-05-18T02:57:07Z"},{"alias_kind":"pith_short_12","alias_value":"3I6QPO673K7H","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"3I6QPO673K7HX45I","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"3I6QPO67","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:37f0e697b4cf2dae7db02531156062bd24f30f6d0fca918aed32d3b8c23f0f30","target":"graph","created_at":"2026-05-18T02:57:07Z","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":"We use discrete Morse theory to give a new proof of the Degree Theorem in Auter space A_n. There is a filtration of A_n into subspaces A_{n,k} using the degree of a graph, and the Degree Theorem says that each A_{n,k} is (k-1)-connected. This result is useful, for example to calculate stability bounds for the homology of Aut(F_n). The standard proof of the Degree Theorem is global in nature. Here we give a proof that only uses local considerations, and lends itself more readily to generalization.","authors_text":"Matthew C. B. Zaremsky, Robert McEwen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2009-07-27T14:34:01Z","title":"A combinatorial proof of the Degree Theorem in Auter space"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.4642","kind":"arxiv","version":4},"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":"sha256:11b3b8a5b13a083e8a08c4e5d5953d593a539a5720b91319dc12522f8f7d73f0","target":"record","created_at":"2026-05-18T02:57:07Z","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":"99e01f7861059d6707de2d1b40b5a46fedc861edbeab9ee0ebb931cda92cbf1f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2009-07-27T14:34:01Z","title_canon_sha256":"90f706a099ece2c5d448c4fb80943ef4bfd05304e6df3dceff11454728f2b5bb"},"schema_version":"1.0","source":{"id":"0907.4642","kind":"arxiv","version":4}},"canonical_sha256":"da3d07bbdfdabe7bf3a85de57e52b98b193b2e2ca15a005b82e5b403c91288cb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"da3d07bbdfdabe7bf3a85de57e52b98b193b2e2ca15a005b82e5b403c91288cb","first_computed_at":"2026-05-18T02:57:07.118398Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:57:07.118398Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XpV3sgZA/ZLNhbXZughtIDg2wqJvPrvsQtpn19Pk1xmt/TbcoeAnNcBYhe+TTpHcwGavoo7c9YAGLE7WJVaGBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:57:07.118882Z","signed_message":"canonical_sha256_bytes"},"source_id":"0907.4642","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:11b3b8a5b13a083e8a08c4e5d5953d593a539a5720b91319dc12522f8f7d73f0","sha256:37f0e697b4cf2dae7db02531156062bd24f30f6d0fca918aed32d3b8c23f0f30"],"state_sha256":"c2a8957e337e2cb1fc489d9a9434fe444e76e4d6ceaa5b2206a22b5e00d3c15d"}