{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:LULLAG44VMDTSQNUKO5ZXRAEKY","short_pith_number":"pith:LULLAG44","canonical_record":{"source":{"id":"2607.15991","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2026-07-17T14:28:11Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"631021400d2eb9846c5f6a56def1cf0b9b417f36f5193b91b46ba29923406bb6","abstract_canon_sha256":"ded3217dfbe903f10db950fdec8fb4dbab3f0c1bd89a8f9f90f19aa3d5d469ce"},"schema_version":"1.0"},"canonical_sha256":"5d16b01b9cab073941b453bb9bc404561279be0ac8cf794f13614ca387377358","source":{"kind":"arxiv","id":"2607.15991","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.15991","created_at":"2026-07-20T02:19:24Z"},{"alias_kind":"arxiv_version","alias_value":"2607.15991v1","created_at":"2026-07-20T02:19:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.15991","created_at":"2026-07-20T02:19:24Z"},{"alias_kind":"pith_short_12","alias_value":"LULLAG44VMDT","created_at":"2026-07-20T02:19:24Z"},{"alias_kind":"pith_short_16","alias_value":"LULLAG44VMDTSQNU","created_at":"2026-07-20T02:19:24Z"},{"alias_kind":"pith_short_8","alias_value":"LULLAG44","created_at":"2026-07-20T02:19:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:LULLAG44VMDTSQNUKO5ZXRAEKY","target":"record","payload":{"canonical_record":{"source":{"id":"2607.15991","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2026-07-17T14:28:11Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"631021400d2eb9846c5f6a56def1cf0b9b417f36f5193b91b46ba29923406bb6","abstract_canon_sha256":"ded3217dfbe903f10db950fdec8fb4dbab3f0c1bd89a8f9f90f19aa3d5d469ce"},"schema_version":"1.0"},"canonical_sha256":"5d16b01b9cab073941b453bb9bc404561279be0ac8cf794f13614ca387377358","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-20T02:19:24.509006Z","signature_b64":"AnCkU1RFl842kmAkFXgFN7K5PvudEvYg1mtVkeTCO+mW7qh0cn72jk8yDACX3qFmvuuzNUglgS13+1FH7YqMDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5d16b01b9cab073941b453bb9bc404561279be0ac8cf794f13614ca387377358","last_reissued_at":"2026-07-20T02:19:24.508219Z","signature_status":"signed_v1","first_computed_at":"2026-07-20T02:19:24.508219Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.15991","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":"2026-07-20T02:19:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sCLkhMfKK63msE0hOcX55ou5FG607CcPJFABiRrQlnCEAt+RKGT0P1yK98b/lJS9ZbKac/cD4APTRofgAFU5CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T15:02:18.608826Z"},"content_sha256":"f16e8a71f6d0dbfa990b6457b9f252666e59c5e0dd2289da079bc2d81a326445","schema_version":"1.0","event_id":"sha256:f16e8a71f6d0dbfa990b6457b9f252666e59c5e0dd2289da079bc2d81a326445"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:LULLAG44VMDTSQNUKO5ZXRAEKY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Wide subcategories and brick-finiteness for length categories","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.RT","authors_text":"Francesco Sentieri","submitted_at":"2026-07-17T14:28:11Z","abstract_excerpt":"We extend to abelian length categories of finite rank a characterisation of brick-finiteness known for finite-dimensional algebras. We prove that such a category is brick-finite if and only if every torsion class is generated by a wide subcategory and every torsionfree class is cogenerated by a wide subcategory. The proof is based on an exchange relation for brick labels across wide intervals which is a shadow of the mutation of simple minded collections. As a corollary,we extend the validity of the first Brick Brauer-Thrall conjecture to this setting."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.15991","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.15991/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-20T02:19:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uYAieVhYuFBK0h1IYUDx+eM7ARxEeNusz1uMyulJVM49+unezV4+An1uoxjeB3gb9b6KtJV0kkJzi8VWC+7iDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T15:02:18.609804Z"},"content_sha256":"7be239e36f8f005cd82f7d4b8276b2f280bc67f16cb9b828076cb2f6d1ec026c","schema_version":"1.0","event_id":"sha256:7be239e36f8f005cd82f7d4b8276b2f280bc67f16cb9b828076cb2f6d1ec026c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LULLAG44VMDTSQNUKO5ZXRAEKY/bundle.json","state_url":"https://pith.science/pith/LULLAG44VMDTSQNUKO5ZXRAEKY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LULLAG44VMDTSQNUKO5ZXRAEKY/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":"2026-08-20T15:02:18Z","links":{"resolver":"https://pith.science/pith/LULLAG44VMDTSQNUKO5ZXRAEKY","bundle":"https://pith.science/pith/LULLAG44VMDTSQNUKO5ZXRAEKY/bundle.json","state":"https://pith.science/pith/LULLAG44VMDTSQNUKO5ZXRAEKY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LULLAG44VMDTSQNUKO5ZXRAEKY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:LULLAG44VMDTSQNUKO5ZXRAEKY","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":"ded3217dfbe903f10db950fdec8fb4dbab3f0c1bd89a8f9f90f19aa3d5d469ce","cross_cats_sorted":["math.CT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2026-07-17T14:28:11Z","title_canon_sha256":"631021400d2eb9846c5f6a56def1cf0b9b417f36f5193b91b46ba29923406bb6"},"schema_version":"1.0","source":{"id":"2607.15991","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.15991","created_at":"2026-07-20T02:19:24Z"},{"alias_kind":"arxiv_version","alias_value":"2607.15991v1","created_at":"2026-07-20T02:19:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.15991","created_at":"2026-07-20T02:19:24Z"},{"alias_kind":"pith_short_12","alias_value":"LULLAG44VMDT","created_at":"2026-07-20T02:19:24Z"},{"alias_kind":"pith_short_16","alias_value":"LULLAG44VMDTSQNU","created_at":"2026-07-20T02:19:24Z"},{"alias_kind":"pith_short_8","alias_value":"LULLAG44","created_at":"2026-07-20T02:19:24Z"}],"graph_snapshots":[{"event_id":"sha256:7be239e36f8f005cd82f7d4b8276b2f280bc67f16cb9b828076cb2f6d1ec026c","target":"graph","created_at":"2026-07-20T02:19:24Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2607.15991/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We extend to abelian length categories of finite rank a characterisation of brick-finiteness known for finite-dimensional algebras. We prove that such a category is brick-finite if and only if every torsion class is generated by a wide subcategory and every torsionfree class is cogenerated by a wide subcategory. The proof is based on an exchange relation for brick labels across wide intervals which is a shadow of the mutation of simple minded collections. As a corollary,we extend the validity of the first Brick Brauer-Thrall conjecture to this setting.","authors_text":"Francesco Sentieri","cross_cats":["math.CT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2026-07-17T14:28:11Z","title":"Wide subcategories and brick-finiteness for length categories"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.15991","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:f16e8a71f6d0dbfa990b6457b9f252666e59c5e0dd2289da079bc2d81a326445","target":"record","created_at":"2026-07-20T02:19:24Z","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":"ded3217dfbe903f10db950fdec8fb4dbab3f0c1bd89a8f9f90f19aa3d5d469ce","cross_cats_sorted":["math.CT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2026-07-17T14:28:11Z","title_canon_sha256":"631021400d2eb9846c5f6a56def1cf0b9b417f36f5193b91b46ba29923406bb6"},"schema_version":"1.0","source":{"id":"2607.15991","kind":"arxiv","version":1}},"canonical_sha256":"5d16b01b9cab073941b453bb9bc404561279be0ac8cf794f13614ca387377358","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5d16b01b9cab073941b453bb9bc404561279be0ac8cf794f13614ca387377358","first_computed_at":"2026-07-20T02:19:24.508219Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-20T02:19:24.508219Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AnCkU1RFl842kmAkFXgFN7K5PvudEvYg1mtVkeTCO+mW7qh0cn72jk8yDACX3qFmvuuzNUglgS13+1FH7YqMDA==","signature_status":"signed_v1","signed_at":"2026-07-20T02:19:24.509006Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.15991","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f16e8a71f6d0dbfa990b6457b9f252666e59c5e0dd2289da079bc2d81a326445","sha256:7be239e36f8f005cd82f7d4b8276b2f280bc67f16cb9b828076cb2f6d1ec026c"],"state_sha256":"31abfa70f3f903cd4eff0fd3269a397a3554a2d2bf5ca4f4498fdd9f96c25663"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eSg/Bz2WGNJRhvMAb8RWtqn5ttpUn2ykkfmOnGh0/HS/RtosUrZM2JxqYlkikW2v+zudeNC7P4akt5Bmoy/aDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-20T15:02:18.615879Z","bundle_sha256":"305d9c0d82a1197b2ecae75f88adab5aa247facfd0a18b779f147c6bf13bdcdf"}}