{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:B3PEMZD2CLHQ56M3VNFPWC4IZD","short_pith_number":"pith:B3PEMZD2","canonical_record":{"source":{"id":"2602.08875","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-02-09T16:35:34Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"1f378975f1b744aedd7b254b91ec701bd1134aa50589188b62b529befd7348d9","abstract_canon_sha256":"443482095c7b0d022ea58f87ed78d6f9f44f2e744e28545545090a6d709889ed"},"schema_version":"1.0"},"canonical_sha256":"0ede46647a12cf0ef99bab4afb0b88c8f323087198191c5c6de793627695a142","source":{"kind":"arxiv","id":"2602.08875","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2602.08875","created_at":"2026-07-14T02:21:29Z"},{"alias_kind":"arxiv_version","alias_value":"2602.08875v2","created_at":"2026-07-14T02:21:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2602.08875","created_at":"2026-07-14T02:21:29Z"},{"alias_kind":"pith_short_12","alias_value":"B3PEMZD2CLHQ","created_at":"2026-07-14T02:21:29Z"},{"alias_kind":"pith_short_16","alias_value":"B3PEMZD2CLHQ56M3","created_at":"2026-07-14T02:21:29Z"},{"alias_kind":"pith_short_8","alias_value":"B3PEMZD2","created_at":"2026-07-14T02:21:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:B3PEMZD2CLHQ56M3VNFPWC4IZD","target":"record","payload":{"canonical_record":{"source":{"id":"2602.08875","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-02-09T16:35:34Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"1f378975f1b744aedd7b254b91ec701bd1134aa50589188b62b529befd7348d9","abstract_canon_sha256":"443482095c7b0d022ea58f87ed78d6f9f44f2e744e28545545090a6d709889ed"},"schema_version":"1.0"},"canonical_sha256":"0ede46647a12cf0ef99bab4afb0b88c8f323087198191c5c6de793627695a142","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-14T02:21:29.203774Z","signature_b64":"gsY+w+xQmOiqQedGMvHh7jlMLm8tCOMuXA2NBHygSnMAgakHu4vZ/jeV+9eCcZqJRjoEoUV40vlEO0OzB1OABw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0ede46647a12cf0ef99bab4afb0b88c8f323087198191c5c6de793627695a142","last_reissued_at":"2026-07-14T02:21:29.202842Z","signature_status":"signed_v1","first_computed_at":"2026-07-14T02:21:29.202842Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2602.08875","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":"2026-07-14T02:21:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pAanbtlWuqKbdZWP6YuQMDkVxtySQTGDB+y/LbDu9DLo9qajO9Mpd2zqVR9JfiS0YYuGjad++CNeILtCDdUzAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T19:00:19.047931Z"},"content_sha256":"f86102927beda7fb343700dc434b66a1e9ab569b000a6c0e03a6ad97f15c8453","schema_version":"1.0","event_id":"sha256:f86102927beda7fb343700dc434b66a1e9ab569b000a6c0e03a6ad97f15c8453"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:B3PEMZD2CLHQ56M3VNFPWC4IZD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On medial Latin quandles and affine modules","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.GR","authors_text":"Luc Ta","submitted_at":"2026-02-09T16:35:34Z","abstract_excerpt":"In this note, we show that the category of Latin (resp. commutative) medial quandles is equivalent to the category of affine modules over a certain Laurent polynomial ring (resp. the dyadic rationals). As applications, we describe free objects in these categories and obtain a structure theorem for finitely generated medial commutative quandles. We also characterize racks whose duals are commutative. Collectively, this solves two open problems of Bardakov and Elhamdadi (arXiv:2601.07057)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2602.08875","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/2602.08875/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":"2026-07-14T02:21:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TYk1LpgcbY+z0+5W9fyWHUxEG9/F1p2AzAHPJsJbg+G2JQhaZwvNiNOAOQdHwhkd93mse9oORdR2mNAFK+7NAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T19:00:19.048429Z"},"content_sha256":"08e7d07efcb6d971a491ad7e3951354bf36cda0205d888f3ad97e39f7079df62","schema_version":"1.0","event_id":"sha256:08e7d07efcb6d971a491ad7e3951354bf36cda0205d888f3ad97e39f7079df62"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/B3PEMZD2CLHQ56M3VNFPWC4IZD/bundle.json","state_url":"https://pith.science/pith/B3PEMZD2CLHQ56M3VNFPWC4IZD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/B3PEMZD2CLHQ56M3VNFPWC4IZD/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":"2026-08-08T19:00:19Z","links":{"resolver":"https://pith.science/pith/B3PEMZD2CLHQ56M3VNFPWC4IZD","bundle":"https://pith.science/pith/B3PEMZD2CLHQ56M3VNFPWC4IZD/bundle.json","state":"https://pith.science/pith/B3PEMZD2CLHQ56M3VNFPWC4IZD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/B3PEMZD2CLHQ56M3VNFPWC4IZD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:B3PEMZD2CLHQ56M3VNFPWC4IZD","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":"443482095c7b0d022ea58f87ed78d6f9f44f2e744e28545545090a6d709889ed","cross_cats_sorted":["math.RA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-02-09T16:35:34Z","title_canon_sha256":"1f378975f1b744aedd7b254b91ec701bd1134aa50589188b62b529befd7348d9"},"schema_version":"1.0","source":{"id":"2602.08875","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2602.08875","created_at":"2026-07-14T02:21:29Z"},{"alias_kind":"arxiv_version","alias_value":"2602.08875v2","created_at":"2026-07-14T02:21:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2602.08875","created_at":"2026-07-14T02:21:29Z"},{"alias_kind":"pith_short_12","alias_value":"B3PEMZD2CLHQ","created_at":"2026-07-14T02:21:29Z"},{"alias_kind":"pith_short_16","alias_value":"B3PEMZD2CLHQ56M3","created_at":"2026-07-14T02:21:29Z"},{"alias_kind":"pith_short_8","alias_value":"B3PEMZD2","created_at":"2026-07-14T02:21:29Z"}],"graph_snapshots":[{"event_id":"sha256:08e7d07efcb6d971a491ad7e3951354bf36cda0205d888f3ad97e39f7079df62","target":"graph","created_at":"2026-07-14T02:21:29Z","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/2602.08875/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this note, we show that the category of Latin (resp. commutative) medial quandles is equivalent to the category of affine modules over a certain Laurent polynomial ring (resp. the dyadic rationals). As applications, we describe free objects in these categories and obtain a structure theorem for finitely generated medial commutative quandles. We also characterize racks whose duals are commutative. Collectively, this solves two open problems of Bardakov and Elhamdadi (arXiv:2601.07057).","authors_text":"Luc Ta","cross_cats":["math.RA"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-02-09T16:35:34Z","title":"On medial Latin quandles and affine modules"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2602.08875","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:f86102927beda7fb343700dc434b66a1e9ab569b000a6c0e03a6ad97f15c8453","target":"record","created_at":"2026-07-14T02:21:29Z","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":"443482095c7b0d022ea58f87ed78d6f9f44f2e744e28545545090a6d709889ed","cross_cats_sorted":["math.RA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-02-09T16:35:34Z","title_canon_sha256":"1f378975f1b744aedd7b254b91ec701bd1134aa50589188b62b529befd7348d9"},"schema_version":"1.0","source":{"id":"2602.08875","kind":"arxiv","version":2}},"canonical_sha256":"0ede46647a12cf0ef99bab4afb0b88c8f323087198191c5c6de793627695a142","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0ede46647a12cf0ef99bab4afb0b88c8f323087198191c5c6de793627695a142","first_computed_at":"2026-07-14T02:21:29.202842Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-14T02:21:29.202842Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gsY+w+xQmOiqQedGMvHh7jlMLm8tCOMuXA2NBHygSnMAgakHu4vZ/jeV+9eCcZqJRjoEoUV40vlEO0OzB1OABw==","signature_status":"signed_v1","signed_at":"2026-07-14T02:21:29.203774Z","signed_message":"canonical_sha256_bytes"},"source_id":"2602.08875","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f86102927beda7fb343700dc434b66a1e9ab569b000a6c0e03a6ad97f15c8453","sha256:08e7d07efcb6d971a491ad7e3951354bf36cda0205d888f3ad97e39f7079df62"],"state_sha256":"13d20ee27cfbc5311c278ff33a39fba04f0dd0c45897f17337c52adf8865eefd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bOXDHe1FDeDmGEfARwL7B+6WPd9FszNPVcTrQ1/6LTX2wl6/3q9T6pmdZOHFCnpsJnzVqucA+ZeNba1CEA+mCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T19:00:19.052176Z","bundle_sha256":"18b59a33db35ecde731181c5e64ab35c8687738612bf2acc59add5374139b368"}}