{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:TVQ2OE4CSRVUIF6DUKDCLSYWEJ","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":"20525a883741a7aa6e8303eb804411d9df864dc2abbf15cdffc6a4df78185f29","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RA","submitted_at":"2026-08-10T05:53:51Z","title_canon_sha256":"d1dbd3410bcc01a96db2d4825d6af55b216053863a74d94eb80657972dbc5be6"},"schema_version":"1.0","source":{"id":"2608.09149","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.09149","created_at":"2026-08-11T02:21:55Z"},{"alias_kind":"arxiv_version","alias_value":"2608.09149v1","created_at":"2026-08-11T02:21:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.09149","created_at":"2026-08-11T02:21:55Z"},{"alias_kind":"pith_short_12","alias_value":"TVQ2OE4CSRVU","created_at":"2026-08-11T02:21:55Z"},{"alias_kind":"pith_short_16","alias_value":"TVQ2OE4CSRVUIF6D","created_at":"2026-08-11T02:21:55Z"},{"alias_kind":"pith_short_8","alias_value":"TVQ2OE4C","created_at":"2026-08-11T02:21:55Z"}],"graph_snapshots":[{"event_id":"sha256:0913301e7c5dfa047a5d911d7a1e87908c640c3dff6e84435831224b8e31c5b1","target":"graph","created_at":"2026-08-11T02:21:55Z","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/2608.09149/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We investigate core quandles and idempotents in quandle rings of core quandles. We answer several questions on the rank of core quandles and nontrivial idempotents in quandle rings. We also present solutions to two questions raised in a recent paper about non-trivial idempotents in quandle rings $\\mathbb{Z}[R_5]$ and $\\mathbb{Z}[C_5]$, where $\\mathbb{Z}$, $R_5$ and $C_5$ are the ring of integers, the dihedral quandle of order 5, and the commutative quandle of order 5, respectively. We then study units in extended quandle rings of a trivial quandle and the Joyce quandle, and nilpotent elements ","authors_text":"Bhitali Kousik, Gregory Churchill, Indu Rasika Churchill, Neranga Fernando","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RA","submitted_at":"2026-08-10T05:53:51Z","title":"A further study of quandles and quandle rings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.09149","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":"sha256:01a425c4ccdb1d93384b7f23c5d404fbe5d55912d09d96d8c60aacb781f59f07","target":"record","created_at":"2026-08-11T02:21:55Z","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":"20525a883741a7aa6e8303eb804411d9df864dc2abbf15cdffc6a4df78185f29","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RA","submitted_at":"2026-08-10T05:53:51Z","title_canon_sha256":"d1dbd3410bcc01a96db2d4825d6af55b216053863a74d94eb80657972dbc5be6"},"schema_version":"1.0","source":{"id":"2608.09149","kind":"arxiv","version":1}},"canonical_sha256":"9d61a71382946b4417c3a28625cb1622758db6fffe7a57f49ac8d7d6f6b81bab","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9d61a71382946b4417c3a28625cb1622758db6fffe7a57f49ac8d7d6f6b81bab","first_computed_at":"2026-08-11T02:21:55.259305Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-11T02:21:55.259305Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MH6rXfKtBERNhM9WM9Od7bq3lT1yy0Ifu9XGyeAKzJKFFLrYXxI1ridijD/iyFIKgfqzDAbetLODbEn7nMUkBg==","signature_status":"signed_v1","signed_at":"2026-08-11T02:21:55.260945Z","signed_message":"canonical_sha256_bytes"},"source_id":"2608.09149","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:01a425c4ccdb1d93384b7f23c5d404fbe5d55912d09d96d8c60aacb781f59f07","sha256:0913301e7c5dfa047a5d911d7a1e87908c640c3dff6e84435831224b8e31c5b1"],"state_sha256":"379d60850e7b6adfda5632bc2ba8aabcb4dfa6faf354f4f6bac3710fcd6ce3d2"}