{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:ZFZTGBO5PMN2KQAOHODKVN7K7Y","short_pith_number":"pith:ZFZTGBO5","canonical_record":{"source":{"id":"2606.11321","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-09T18:03:12Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"f3d34abc64fe74d0d544effcb02167cb92e966a1cb7cb480aba789a18a21bc22","abstract_canon_sha256":"75548e7bc54f0f0685c76273465bac365fdc618f03bdf095a40ae0c5ca19e54f"},"schema_version":"1.0"},"canonical_sha256":"c9733305dd7b1ba5400e3b86aab7eafe068d5e9184207e8ac3a8d9926c97e9ae","source":{"kind":"arxiv","id":"2606.11321","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.11321","created_at":"2026-06-11T00:08:19Z"},{"alias_kind":"arxiv_version","alias_value":"2606.11321v1","created_at":"2026-06-11T00:08:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.11321","created_at":"2026-06-11T00:08:19Z"},{"alias_kind":"pith_short_12","alias_value":"ZFZTGBO5PMN2","created_at":"2026-06-11T00:08:19Z"},{"alias_kind":"pith_short_16","alias_value":"ZFZTGBO5PMN2KQAO","created_at":"2026-06-11T00:08:19Z"},{"alias_kind":"pith_short_8","alias_value":"ZFZTGBO5","created_at":"2026-06-11T00:08:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:ZFZTGBO5PMN2KQAOHODKVN7K7Y","target":"record","payload":{"canonical_record":{"source":{"id":"2606.11321","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-09T18:03:12Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"f3d34abc64fe74d0d544effcb02167cb92e966a1cb7cb480aba789a18a21bc22","abstract_canon_sha256":"75548e7bc54f0f0685c76273465bac365fdc618f03bdf095a40ae0c5ca19e54f"},"schema_version":"1.0"},"canonical_sha256":"c9733305dd7b1ba5400e3b86aab7eafe068d5e9184207e8ac3a8d9926c97e9ae","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-11T00:08:19.262908Z","signature_b64":"3xJDSFS7cuTYz2dHqURnvTH8vtGP4Sz3EAjGkMI7pZtKnLuuFHAWapoMEllpbdTbDdPVcA8vdmRKVetJqOw1Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c9733305dd7b1ba5400e3b86aab7eafe068d5e9184207e8ac3a8d9926c97e9ae","last_reissued_at":"2026-06-11T00:08:19.262165Z","signature_status":"signed_v1","first_computed_at":"2026-06-11T00:08:19.262165Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.11321","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":"2026-06-11T00:08:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ffCqCw6UKB7ke9erugw2hrApJwFSOYDMvf7nuz+5tOdbk7+gcnihoS/S7HOHeiOo6AxU8V6B5zU3XSiXSvFtAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T18:55:48.701122Z"},"content_sha256":"20a4f7ede32614302068ec516955a4e1caa0c5821a41bfd53663511ed5d1ae95","schema_version":"1.0","event_id":"sha256:20a4f7ede32614302068ec516955a4e1caa0c5821a41bfd53663511ed5d1ae95"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:ZFZTGBO5PMN2KQAOHODKVN7K7Y","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Terwilliger $\\mathbb{F}$-algebras of factorial association schemes II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.CO","authors_text":"Yu Jiang","submitted_at":"2026-06-09T18:03:12Z","abstract_excerpt":"The Terwilliger algebras of association schemes over an arbitrary field $\\mathbb{F}$ were called the Terwilliger $\\mathbb{F}$-algebras of association schemes in [10]. In [7], He and Jiang studied the Terwilliger $\\mathbb{F}$-algebras of factorial association schemes. In this paper, we continue studying the Terwilliger $\\mathbb{F}$-algebras of factorial association schemes. We get all block idempotents of the Terwilliger $\\mathbb{F}$-algebras of factorial association schemes. We get the $\\mathbb{F}$-dimensions, the centers, the Jacobson radicals of the block algebras of the Terwilliger $\\mathbb"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11321","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/2606.11321/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":"2026-06-11T00:08:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zwmZE8oTFbIaF1mO0z91FTJtyrT30End7e9DpyxSGG3qcQskG32zlwnATXY21w6RnbtBpeOorkd+qbUs/OEOCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T18:55:48.701509Z"},"content_sha256":"dea09d7c52f83ca248e54ab257a01ef267caf51dbae6c1835dfc73c608c84114","schema_version":"1.0","event_id":"sha256:dea09d7c52f83ca248e54ab257a01ef267caf51dbae6c1835dfc73c608c84114"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZFZTGBO5PMN2KQAOHODKVN7K7Y/bundle.json","state_url":"https://pith.science/pith/ZFZTGBO5PMN2KQAOHODKVN7K7Y/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZFZTGBO5PMN2KQAOHODKVN7K7Y/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":"2026-06-30T18:55:48Z","links":{"resolver":"https://pith.science/pith/ZFZTGBO5PMN2KQAOHODKVN7K7Y","bundle":"https://pith.science/pith/ZFZTGBO5PMN2KQAOHODKVN7K7Y/bundle.json","state":"https://pith.science/pith/ZFZTGBO5PMN2KQAOHODKVN7K7Y/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZFZTGBO5PMN2KQAOHODKVN7K7Y/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:ZFZTGBO5PMN2KQAOHODKVN7K7Y","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":"75548e7bc54f0f0685c76273465bac365fdc618f03bdf095a40ae0c5ca19e54f","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-09T18:03:12Z","title_canon_sha256":"f3d34abc64fe74d0d544effcb02167cb92e966a1cb7cb480aba789a18a21bc22"},"schema_version":"1.0","source":{"id":"2606.11321","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.11321","created_at":"2026-06-11T00:08:19Z"},{"alias_kind":"arxiv_version","alias_value":"2606.11321v1","created_at":"2026-06-11T00:08:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.11321","created_at":"2026-06-11T00:08:19Z"},{"alias_kind":"pith_short_12","alias_value":"ZFZTGBO5PMN2","created_at":"2026-06-11T00:08:19Z"},{"alias_kind":"pith_short_16","alias_value":"ZFZTGBO5PMN2KQAO","created_at":"2026-06-11T00:08:19Z"},{"alias_kind":"pith_short_8","alias_value":"ZFZTGBO5","created_at":"2026-06-11T00:08:19Z"}],"graph_snapshots":[{"event_id":"sha256:dea09d7c52f83ca248e54ab257a01ef267caf51dbae6c1835dfc73c608c84114","target":"graph","created_at":"2026-06-11T00:08:19Z","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/2606.11321/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The Terwilliger algebras of association schemes over an arbitrary field $\\mathbb{F}$ were called the Terwilliger $\\mathbb{F}$-algebras of association schemes in [10]. In [7], He and Jiang studied the Terwilliger $\\mathbb{F}$-algebras of factorial association schemes. In this paper, we continue studying the Terwilliger $\\mathbb{F}$-algebras of factorial association schemes. We get all block idempotents of the Terwilliger $\\mathbb{F}$-algebras of factorial association schemes. We get the $\\mathbb{F}$-dimensions, the centers, the Jacobson radicals of the block algebras of the Terwilliger $\\mathbb","authors_text":"Yu Jiang","cross_cats":["math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-09T18:03:12Z","title":"On Terwilliger $\\mathbb{F}$-algebras of factorial association schemes II"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11321","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":"sha256:20a4f7ede32614302068ec516955a4e1caa0c5821a41bfd53663511ed5d1ae95","target":"record","created_at":"2026-06-11T00:08:19Z","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":"75548e7bc54f0f0685c76273465bac365fdc618f03bdf095a40ae0c5ca19e54f","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-09T18:03:12Z","title_canon_sha256":"f3d34abc64fe74d0d544effcb02167cb92e966a1cb7cb480aba789a18a21bc22"},"schema_version":"1.0","source":{"id":"2606.11321","kind":"arxiv","version":1}},"canonical_sha256":"c9733305dd7b1ba5400e3b86aab7eafe068d5e9184207e8ac3a8d9926c97e9ae","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c9733305dd7b1ba5400e3b86aab7eafe068d5e9184207e8ac3a8d9926c97e9ae","first_computed_at":"2026-06-11T00:08:19.262165Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-11T00:08:19.262165Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3xJDSFS7cuTYz2dHqURnvTH8vtGP4Sz3EAjGkMI7pZtKnLuuFHAWapoMEllpbdTbDdPVcA8vdmRKVetJqOw1Aw==","signature_status":"signed_v1","signed_at":"2026-06-11T00:08:19.262908Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.11321","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:20a4f7ede32614302068ec516955a4e1caa0c5821a41bfd53663511ed5d1ae95","sha256:dea09d7c52f83ca248e54ab257a01ef267caf51dbae6c1835dfc73c608c84114"],"state_sha256":"f08e8b526304a1b639718548e4fce692da19dda2fb5783db492440de06d00a36"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3oEbWoJGIr75wuxcHKN0VkUtWYsHVokoODOkA+gyJTVt2q379kLh3yB492X5bM6ZPCqWTJHSSVJTPg6E1rQ/AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-30T18:55:48.703504Z","bundle_sha256":"196e4fb86459eb76ea89b34d1f35fd1261239a1c0df710665b1132552adf7fc6"}}