{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:U43S4WPQHFBFP6BZ74KRCKLDCU","short_pith_number":"pith:U43S4WPQ","canonical_record":{"source":{"id":"2506.13602","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2025-06-16T15:25:46Z","cross_cats_sorted":["math.CO","math.RA"],"title_canon_sha256":"adfb44b996744ce5df54703bfdb0fb13dcfd29aa6f09f6d8e1dbf84aefa97be7","abstract_canon_sha256":"75e2fba495fcb9b097cbd751e746e851b1c38d7cf3a1bbbbdbfcb52740d7c149"},"schema_version":"1.0"},"canonical_sha256":"a7372e59f0394257f839ff15112963151a2dd7d13d105006366ffa1c4a3699de","source":{"kind":"arxiv","id":"2506.13602","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.13602","created_at":"2026-07-05T12:00:02Z"},{"alias_kind":"arxiv_version","alias_value":"2506.13602v2","created_at":"2026-07-05T12:00:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.13602","created_at":"2026-07-05T12:00:02Z"},{"alias_kind":"pith_short_12","alias_value":"U43S4WPQHFBF","created_at":"2026-07-05T12:00:02Z"},{"alias_kind":"pith_short_16","alias_value":"U43S4WPQHFBFP6BZ","created_at":"2026-07-05T12:00:02Z"},{"alias_kind":"pith_short_8","alias_value":"U43S4WPQ","created_at":"2026-07-05T12:00:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:U43S4WPQHFBFP6BZ74KRCKLDCU","target":"record","payload":{"canonical_record":{"source":{"id":"2506.13602","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2025-06-16T15:25:46Z","cross_cats_sorted":["math.CO","math.RA"],"title_canon_sha256":"adfb44b996744ce5df54703bfdb0fb13dcfd29aa6f09f6d8e1dbf84aefa97be7","abstract_canon_sha256":"75e2fba495fcb9b097cbd751e746e851b1c38d7cf3a1bbbbdbfcb52740d7c149"},"schema_version":"1.0"},"canonical_sha256":"a7372e59f0394257f839ff15112963151a2dd7d13d105006366ffa1c4a3699de","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:00:02.430863Z","signature_b64":"SGYOv6iMGoJB0wAoPmt1Z8U4UUBSDDyhrVEUR7XbkMezPVb8F1Ae7Sw1yJDWk4DeGTw+M+UdGkyi+61xokqRAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a7372e59f0394257f839ff15112963151a2dd7d13d105006366ffa1c4a3699de","last_reissued_at":"2026-07-05T12:00:02.430364Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:00:02.430364Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2506.13602","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T12:00:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oywyN/tTmGBZLGuEt+Hfn9hpr0EvvXLm9wCh7ppOMkxQEy5Ne4fdJf037eQ6mV+9kAXhvMdxCaS9TZF5+CpxCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T06:06:37.627160Z"},"content_sha256":"da155ba0896cd1c7420ba240865ef4b4349c7469c6091c85bd329b4a65d564b9","schema_version":"1.0","event_id":"sha256:da155ba0896cd1c7420ba240865ef4b4349c7469c6091c85bd329b4a65d564b9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:U43S4WPQHFBFP6BZ74KRCKLDCU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Brick-splitting Torsion Pairs and Left Modularity","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO","math.RA"],"primary_cat":"math.RT","authors_text":"Charles Paquette, Kaveh Mousavand, Osamu Iyama, Sota Asai","submitted_at":"2025-06-16T15:25:46Z","abstract_excerpt":"We introduce the notion of brick-splitting torsion pairs as a modern analogue and generalization of the classical notion of splitting torsion pairs. A torsion pair is called brick-splitting if any given brick is either torsion or torsion-free with respect to that torsion pair. After giving some properties of these pairs, we fully characterize them in terms of some lattice-theoretical properties, including left modularity. This leads to the notion of brick-directed algebras, which are those for which there does not exist any cycle of non-zero non-isomorphisms between bricks. This class of algeb"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.13602","kind":"arxiv","version":2},"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/2506.13602/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":"2026-07-05T12:00:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BVh8dlMq1BV94VSiRyY+OWsbNDpxJve48E+WA4BEfLhrTgI6wDd03Nm5QXOnu7hhMJEa7UdWCHYA1HAkUexyCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T06:06:37.627704Z"},"content_sha256":"980e05e1c18a8e6e75442bb217dcf1cbb95bca4d13982971f85f01e6d48d8194","schema_version":"1.0","event_id":"sha256:980e05e1c18a8e6e75442bb217dcf1cbb95bca4d13982971f85f01e6d48d8194"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/U43S4WPQHFBFP6BZ74KRCKLDCU/bundle.json","state_url":"https://pith.science/pith/U43S4WPQHFBFP6BZ74KRCKLDCU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/U43S4WPQHFBFP6BZ74KRCKLDCU/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":"2026-08-19T06:06:37Z","links":{"resolver":"https://pith.science/pith/U43S4WPQHFBFP6BZ74KRCKLDCU","bundle":"https://pith.science/pith/U43S4WPQHFBFP6BZ74KRCKLDCU/bundle.json","state":"https://pith.science/pith/U43S4WPQHFBFP6BZ74KRCKLDCU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/U43S4WPQHFBFP6BZ74KRCKLDCU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:U43S4WPQHFBFP6BZ74KRCKLDCU","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":"75e2fba495fcb9b097cbd751e746e851b1c38d7cf3a1bbbbdbfcb52740d7c149","cross_cats_sorted":["math.CO","math.RA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2025-06-16T15:25:46Z","title_canon_sha256":"adfb44b996744ce5df54703bfdb0fb13dcfd29aa6f09f6d8e1dbf84aefa97be7"},"schema_version":"1.0","source":{"id":"2506.13602","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.13602","created_at":"2026-07-05T12:00:02Z"},{"alias_kind":"arxiv_version","alias_value":"2506.13602v2","created_at":"2026-07-05T12:00:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.13602","created_at":"2026-07-05T12:00:02Z"},{"alias_kind":"pith_short_12","alias_value":"U43S4WPQHFBF","created_at":"2026-07-05T12:00:02Z"},{"alias_kind":"pith_short_16","alias_value":"U43S4WPQHFBFP6BZ","created_at":"2026-07-05T12:00:02Z"},{"alias_kind":"pith_short_8","alias_value":"U43S4WPQ","created_at":"2026-07-05T12:00:02Z"}],"graph_snapshots":[{"event_id":"sha256:980e05e1c18a8e6e75442bb217dcf1cbb95bca4d13982971f85f01e6d48d8194","target":"graph","created_at":"2026-07-05T12:00:02Z","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/2506.13602/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We introduce the notion of brick-splitting torsion pairs as a modern analogue and generalization of the classical notion of splitting torsion pairs. A torsion pair is called brick-splitting if any given brick is either torsion or torsion-free with respect to that torsion pair. After giving some properties of these pairs, we fully characterize them in terms of some lattice-theoretical properties, including left modularity. This leads to the notion of brick-directed algebras, which are those for which there does not exist any cycle of non-zero non-isomorphisms between bricks. This class of algeb","authors_text":"Charles Paquette, Kaveh Mousavand, Osamu Iyama, Sota Asai","cross_cats":["math.CO","math.RA"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2025-06-16T15:25:46Z","title":"Brick-splitting Torsion Pairs and Left Modularity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.13602","kind":"arxiv","version":2},"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:da155ba0896cd1c7420ba240865ef4b4349c7469c6091c85bd329b4a65d564b9","target":"record","created_at":"2026-07-05T12:00:02Z","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":"75e2fba495fcb9b097cbd751e746e851b1c38d7cf3a1bbbbdbfcb52740d7c149","cross_cats_sorted":["math.CO","math.RA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2025-06-16T15:25:46Z","title_canon_sha256":"adfb44b996744ce5df54703bfdb0fb13dcfd29aa6f09f6d8e1dbf84aefa97be7"},"schema_version":"1.0","source":{"id":"2506.13602","kind":"arxiv","version":2}},"canonical_sha256":"a7372e59f0394257f839ff15112963151a2dd7d13d105006366ffa1c4a3699de","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a7372e59f0394257f839ff15112963151a2dd7d13d105006366ffa1c4a3699de","first_computed_at":"2026-07-05T12:00:02.430364Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T12:00:02.430364Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SGYOv6iMGoJB0wAoPmt1Z8U4UUBSDDyhrVEUR7XbkMezPVb8F1Ae7Sw1yJDWk4DeGTw+M+UdGkyi+61xokqRAA==","signature_status":"signed_v1","signed_at":"2026-07-05T12:00:02.430863Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.13602","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:da155ba0896cd1c7420ba240865ef4b4349c7469c6091c85bd329b4a65d564b9","sha256:980e05e1c18a8e6e75442bb217dcf1cbb95bca4d13982971f85f01e6d48d8194"],"state_sha256":"4fc28df2b5666abeb468e60e31f49252d82c5a2a303b08604222ab16322db712"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nri3dHaB/wmGxVBuv69a/2LK68dV6UJXK1Db+EXNH49p3yAD4Ezz3e+dVsXDi8BYEXwUTeNA1FOR5m28z4K8Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-19T06:06:37.633614Z","bundle_sha256":"a5f653704db6f0dab872f0ad37072d097c9d42b31e36f497912928a813cb4b7d"}}