{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:RQF2V72ORUH6ZLROURN3KBAL6I","short_pith_number":"pith:RQF2V72O","canonical_record":{"source":{"id":"2606.27879","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2026-06-26T09:21:35Z","cross_cats_sorted":[],"title_canon_sha256":"9f317a76160f637c697374ea47c0fad259cae8cd6999c23d7948b2ab3d5b83e2","abstract_canon_sha256":"e6dfc49715527572b58b568b3c553bc6bf97806325597659b51af1bae2dce94c"},"schema_version":"1.0"},"canonical_sha256":"8c0baaff4e8d0fecae2ea45bb5040bf2166f376484fac8b4f9dcc8f5df29a0aa","source":{"kind":"arxiv","id":"2606.27879","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.27879","created_at":"2026-06-29T01:14:51Z"},{"alias_kind":"arxiv_version","alias_value":"2606.27879v1","created_at":"2026-06-29T01:14:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.27879","created_at":"2026-06-29T01:14:51Z"},{"alias_kind":"pith_short_12","alias_value":"RQF2V72ORUH6","created_at":"2026-06-29T01:14:51Z"},{"alias_kind":"pith_short_16","alias_value":"RQF2V72ORUH6ZLRO","created_at":"2026-06-29T01:14:51Z"},{"alias_kind":"pith_short_8","alias_value":"RQF2V72O","created_at":"2026-06-29T01:14:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:RQF2V72ORUH6ZLROURN3KBAL6I","target":"record","payload":{"canonical_record":{"source":{"id":"2606.27879","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2026-06-26T09:21:35Z","cross_cats_sorted":[],"title_canon_sha256":"9f317a76160f637c697374ea47c0fad259cae8cd6999c23d7948b2ab3d5b83e2","abstract_canon_sha256":"e6dfc49715527572b58b568b3c553bc6bf97806325597659b51af1bae2dce94c"},"schema_version":"1.0"},"canonical_sha256":"8c0baaff4e8d0fecae2ea45bb5040bf2166f376484fac8b4f9dcc8f5df29a0aa","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-29T01:14:51.471223Z","signature_b64":"+8Ee8vzuunz6IRzhgv/npD+zqP5HBRzBB39kZMcX2zjP5PzWQb2KjChJIf096VWbJszfUyA0EREsyN5RedMsDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8c0baaff4e8d0fecae2ea45bb5040bf2166f376484fac8b4f9dcc8f5df29a0aa","last_reissued_at":"2026-06-29T01:14:51.470820Z","signature_status":"signed_v1","first_computed_at":"2026-06-29T01:14:51.470820Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.27879","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":"2026-06-29T01:14:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yExfjLSqO1vWtYATwBRaIEwS0r5b3HtHofQycM4VvsTYtpH1M4KsIglb+3zZzW9C4VXsF4Zm2RsNo00am1+ECQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-12T19:39:51.122415Z"},"content_sha256":"12c488b0c58a5b3754284fffc57dc3ddc7301157b761b4b58dc2196afa3af081","schema_version":"1.0","event_id":"sha256:12c488b0c58a5b3754284fffc57dc3ddc7301157b761b4b58dc2196afa3af081"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:RQF2V72ORUH6ZLROURN3KBAL6I","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The brick chain complexity of an artin algebra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Claus Michael Ringel","submitted_at":"2026-06-26T09:21:35Z","abstract_excerpt":"We consider the category of finitely generated modules over an artin algebra $A$. It is known that any module $M$ has a brick chain filtration. We say that M has brick chain complexity at most $t$ provided $M$ has a brick chain filtration of length at most $t$. The brick chain complexity of A is by definition the supremum of the brick chain complexity of the indecomposable $A$-modules. The aim of this note is to calculate the brick chain complexity for some algebras. We will exhibit algebras with arbitrarily large brick chain complexity."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.27879","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/2606.27879/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":"2026-06-29T01:14:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cygixEWBKNZJxf6G6tql4qxBxsa3C3Y+CCWrlbyy0ExrJmWYpDUzrq8LA7vQbmN0L+Jb/dXuSClm4F/G1Z54CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-12T19:39:51.122792Z"},"content_sha256":"051ea03c100741d539400a038a11109803852e65d75fc216d7fbf40dffd0f2cb","schema_version":"1.0","event_id":"sha256:051ea03c100741d539400a038a11109803852e65d75fc216d7fbf40dffd0f2cb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RQF2V72ORUH6ZLROURN3KBAL6I/bundle.json","state_url":"https://pith.science/pith/RQF2V72ORUH6ZLROURN3KBAL6I/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RQF2V72ORUH6ZLROURN3KBAL6I/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":"2026-07-12T19:39:51Z","links":{"resolver":"https://pith.science/pith/RQF2V72ORUH6ZLROURN3KBAL6I","bundle":"https://pith.science/pith/RQF2V72ORUH6ZLROURN3KBAL6I/bundle.json","state":"https://pith.science/pith/RQF2V72ORUH6ZLROURN3KBAL6I/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RQF2V72ORUH6ZLROURN3KBAL6I/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:RQF2V72ORUH6ZLROURN3KBAL6I","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":"e6dfc49715527572b58b568b3c553bc6bf97806325597659b51af1bae2dce94c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2026-06-26T09:21:35Z","title_canon_sha256":"9f317a76160f637c697374ea47c0fad259cae8cd6999c23d7948b2ab3d5b83e2"},"schema_version":"1.0","source":{"id":"2606.27879","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.27879","created_at":"2026-06-29T01:14:51Z"},{"alias_kind":"arxiv_version","alias_value":"2606.27879v1","created_at":"2026-06-29T01:14:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.27879","created_at":"2026-06-29T01:14:51Z"},{"alias_kind":"pith_short_12","alias_value":"RQF2V72ORUH6","created_at":"2026-06-29T01:14:51Z"},{"alias_kind":"pith_short_16","alias_value":"RQF2V72ORUH6ZLRO","created_at":"2026-06-29T01:14:51Z"},{"alias_kind":"pith_short_8","alias_value":"RQF2V72O","created_at":"2026-06-29T01:14:51Z"}],"graph_snapshots":[{"event_id":"sha256:051ea03c100741d539400a038a11109803852e65d75fc216d7fbf40dffd0f2cb","target":"graph","created_at":"2026-06-29T01:14:51Z","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/2606.27879/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We consider the category of finitely generated modules over an artin algebra $A$. It is known that any module $M$ has a brick chain filtration. We say that M has brick chain complexity at most $t$ provided $M$ has a brick chain filtration of length at most $t$. The brick chain complexity of A is by definition the supremum of the brick chain complexity of the indecomposable $A$-modules. The aim of this note is to calculate the brick chain complexity for some algebras. We will exhibit algebras with arbitrarily large brick chain complexity.","authors_text":"Claus Michael Ringel","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2026-06-26T09:21:35Z","title":"The brick chain complexity of an artin algebra"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.27879","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":"sha256:12c488b0c58a5b3754284fffc57dc3ddc7301157b761b4b58dc2196afa3af081","target":"record","created_at":"2026-06-29T01:14:51Z","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":"e6dfc49715527572b58b568b3c553bc6bf97806325597659b51af1bae2dce94c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2026-06-26T09:21:35Z","title_canon_sha256":"9f317a76160f637c697374ea47c0fad259cae8cd6999c23d7948b2ab3d5b83e2"},"schema_version":"1.0","source":{"id":"2606.27879","kind":"arxiv","version":1}},"canonical_sha256":"8c0baaff4e8d0fecae2ea45bb5040bf2166f376484fac8b4f9dcc8f5df29a0aa","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8c0baaff4e8d0fecae2ea45bb5040bf2166f376484fac8b4f9dcc8f5df29a0aa","first_computed_at":"2026-06-29T01:14:51.470820Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-29T01:14:51.470820Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+8Ee8vzuunz6IRzhgv/npD+zqP5HBRzBB39kZMcX2zjP5PzWQb2KjChJIf096VWbJszfUyA0EREsyN5RedMsDg==","signature_status":"signed_v1","signed_at":"2026-06-29T01:14:51.471223Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.27879","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:12c488b0c58a5b3754284fffc57dc3ddc7301157b761b4b58dc2196afa3af081","sha256:051ea03c100741d539400a038a11109803852e65d75fc216d7fbf40dffd0f2cb"],"state_sha256":"f3b19f41339f47844e26f848f17f0c8bb88b17fa0b2aeba59bf9587de7875ac8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ExKkqoEvKPghRGL+19nkcqW+uj7bGo1ol9Bx1YcQ8f7gRCXtej/nq+Zj0T2bGS0ed49TbqCFuztmJBPUn0CvCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-12T19:39:51.125470Z","bundle_sha256":"d9fbbe7e9b1080be61ae01a9f368e824de362332f0b8d796cf69215e7f220161"}}