{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:DPADJFVOJYNI63WSGM5Z4J3KBA","short_pith_number":"pith:DPADJFVO","schema_version":"1.0","canonical_sha256":"1bc03496ae4e1a8f6ed2333b9e276a0801faf158e7025efee853a6ebf575c3ba","source":{"kind":"arxiv","id":"1211.3378","version":1},"attestation_state":"computed","paper":{"title":"Reducing systems for very small trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GR","authors_text":"Patrick Reynolds","submitted_at":"2012-11-14T18:40:53Z","abstract_excerpt":"We study very small trees from the point of view of reducing systems of free factors, which are analogues of reducing systems of curves for a surface lamination; a non-trivial, proper free factor $F \\leq \\FN$ reduces $T$ if and only if $F$ acts on some subtree of $T$ with dense orbits. We characterize those trees, called arational, which do not admit a reduction by any free factor: $T$ is arational if and only if either $T$ is free and indecomposable or $T$ is dual to a surface with one boundary component equipped with an arational measured foliation. To complement this result, we establish so"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1211.3378","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-11-14T18:40:53Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"dbe361cb49ce71a1c19bae8a704a332b6cd8c3035bf70b2102de69509aaa95d2","abstract_canon_sha256":"3e7edeeade5e3a76933a56b054f75a1718ecddd639a20c29db306e7d5d6a0cc1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:40:50.042649Z","signature_b64":"BuOaBEKsslppI/Wmi622iZ6AhDjkHD2hpY+k1bsaONV1DYs8RibzKr8IFvxtWnxpeBVWuNoRTZktPTvEC6NyDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1bc03496ae4e1a8f6ed2333b9e276a0801faf158e7025efee853a6ebf575c3ba","last_reissued_at":"2026-05-18T03:40:50.042127Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:40:50.042127Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Reducing systems for very small trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GR","authors_text":"Patrick Reynolds","submitted_at":"2012-11-14T18:40:53Z","abstract_excerpt":"We study very small trees from the point of view of reducing systems of free factors, which are analogues of reducing systems of curves for a surface lamination; a non-trivial, proper free factor $F \\leq \\FN$ reduces $T$ if and only if $F$ acts on some subtree of $T$ with dense orbits. We characterize those trees, called arational, which do not admit a reduction by any free factor: $T$ is arational if and only if either $T$ is free and indecomposable or $T$ is dual to a surface with one boundary component equipped with an arational measured foliation. To complement this result, we establish so"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.3378","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1211.3378","created_at":"2026-05-18T03:40:50.042213+00:00"},{"alias_kind":"arxiv_version","alias_value":"1211.3378v1","created_at":"2026-05-18T03:40:50.042213+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.3378","created_at":"2026-05-18T03:40:50.042213+00:00"},{"alias_kind":"pith_short_12","alias_value":"DPADJFVOJYNI","created_at":"2026-05-18T12:27:04.183437+00:00"},{"alias_kind":"pith_short_16","alias_value":"DPADJFVOJYNI63WS","created_at":"2026-05-18T12:27:04.183437+00:00"},{"alias_kind":"pith_short_8","alias_value":"DPADJFVO","created_at":"2026-05-18T12:27:04.183437+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DPADJFVOJYNI63WSGM5Z4J3KBA","json":"https://pith.science/pith/DPADJFVOJYNI63WSGM5Z4J3KBA.json","graph_json":"https://pith.science/api/pith-number/DPADJFVOJYNI63WSGM5Z4J3KBA/graph.json","events_json":"https://pith.science/api/pith-number/DPADJFVOJYNI63WSGM5Z4J3KBA/events.json","paper":"https://pith.science/paper/DPADJFVO"},"agent_actions":{"view_html":"https://pith.science/pith/DPADJFVOJYNI63WSGM5Z4J3KBA","download_json":"https://pith.science/pith/DPADJFVOJYNI63WSGM5Z4J3KBA.json","view_paper":"https://pith.science/paper/DPADJFVO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1211.3378&json=true","fetch_graph":"https://pith.science/api/pith-number/DPADJFVOJYNI63WSGM5Z4J3KBA/graph.json","fetch_events":"https://pith.science/api/pith-number/DPADJFVOJYNI63WSGM5Z4J3KBA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DPADJFVOJYNI63WSGM5Z4J3KBA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DPADJFVOJYNI63WSGM5Z4J3KBA/action/storage_attestation","attest_author":"https://pith.science/pith/DPADJFVOJYNI63WSGM5Z4J3KBA/action/author_attestation","sign_citation":"https://pith.science/pith/DPADJFVOJYNI63WSGM5Z4J3KBA/action/citation_signature","submit_replication":"https://pith.science/pith/DPADJFVOJYNI63WSGM5Z4J3KBA/action/replication_record"}},"created_at":"2026-05-18T03:40:50.042213+00:00","updated_at":"2026-05-18T03:40:50.042213+00:00"}