{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:5XT3UJEFYDLNAMVNSO3TCMXG7G","short_pith_number":"pith:5XT3UJEF","canonical_record":{"source":{"id":"2607.25246","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2026-07-28T03:40:20Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"c544e4da8c2b3dbea1fceea611e129a1aad13f92f666766dc9074eb17fd752bf","abstract_canon_sha256":"db91ff6ca22a38dd945c6cc71c828a61e0e9ab5f2e1921faf3197bf0695c936a"},"schema_version":"1.0"},"canonical_sha256":"ede7ba2485c0d6d032ad93b73132e6f986bf2bb1e9cda5641edb3ad50bc1bfa9","source":{"kind":"arxiv","id":"2607.25246","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.25246","created_at":"2026-07-29T00:25:10Z"},{"alias_kind":"arxiv_version","alias_value":"2607.25246v1","created_at":"2026-07-29T00:25:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.25246","created_at":"2026-07-29T00:25:10Z"},{"alias_kind":"pith_short_12","alias_value":"5XT3UJEFYDLN","created_at":"2026-07-29T00:25:10Z"},{"alias_kind":"pith_short_16","alias_value":"5XT3UJEFYDLNAMVN","created_at":"2026-07-29T00:25:10Z"},{"alias_kind":"pith_short_8","alias_value":"5XT3UJEF","created_at":"2026-07-29T00:25:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:5XT3UJEFYDLNAMVNSO3TCMXG7G","target":"record","payload":{"canonical_record":{"source":{"id":"2607.25246","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2026-07-28T03:40:20Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"c544e4da8c2b3dbea1fceea611e129a1aad13f92f666766dc9074eb17fd752bf","abstract_canon_sha256":"db91ff6ca22a38dd945c6cc71c828a61e0e9ab5f2e1921faf3197bf0695c936a"},"schema_version":"1.0"},"canonical_sha256":"ede7ba2485c0d6d032ad93b73132e6f986bf2bb1e9cda5641edb3ad50bc1bfa9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-29T00:25:10.599971Z","signature_b64":"lC/Bz8uNki9QFz94mh/Ur1nCEJi5tToRGg4KCYl6BbCrSFJJydI9uU3KE5jah2+ny/DPy3a1FixZi01WLcA+Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ede7ba2485c0d6d032ad93b73132e6f986bf2bb1e9cda5641edb3ad50bc1bfa9","last_reissued_at":"2026-07-29T00:25:10.599113Z","signature_status":"signed_v1","first_computed_at":"2026-07-29T00:25:10.599113Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.25246","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":"2026-07-29T00:25:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"R0NQ/g3Hrbw0CfNqYUJhVPczBx+jOPSIB6mmzQNGbvYbqYh6y1GEo5TK7A4vP6h0CbvI8ijflV8F6P4bPKyvBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T01:49:53.426951Z"},"content_sha256":"7e020efa55f62de9fba913d6b05c5af0c7ed67479c35cd8ac7da6158145cfdf5","schema_version":"1.0","event_id":"sha256:7e020efa55f62de9fba913d6b05c5af0c7ed67479c35cd8ac7da6158145cfdf5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:5XT3UJEFYDLNAMVNSO3TCMXG7G","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Nijenhuis modules and the ring of Nijenhuis operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.RT","authors_text":"Liuqing Dong, Shilong Zhang, Xiao-Song Peng","submitted_at":"2026-07-28T03:40:20Z","abstract_excerpt":"In this paper, we study the Nijenhuis modules of Nijenhuis algebras. The concepts of free, projective, injective, and flat Nijenhuis modules are introduced, and a construction of free Nijenhuis modules is given. The ring of Nijenhuis operators is introduced, and the relationship between Nijenhuis modules and modules over this ring is established by proving an isomorphism of the corresponding categories. Finally, it is proved that the category of Nijenhuis modules has enough projective, injective, and flat objects."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.25246","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/2607.25246/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":"2026-07-29T00:25:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vxWxNL0WSEhRHsIL6Nu3VQZh8O3wGTaCJN2T8zy6pVkZ7kmz0ycgOkeqYNNt35vXoFxR0YNTgq5UU8uyHgSrDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T01:49:53.427446Z"},"content_sha256":"f70d30be43e006c4caea874861ff82f93abc99bd230e6aeed27d74ca131c9e9f","schema_version":"1.0","event_id":"sha256:f70d30be43e006c4caea874861ff82f93abc99bd230e6aeed27d74ca131c9e9f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5XT3UJEFYDLNAMVNSO3TCMXG7G/bundle.json","state_url":"https://pith.science/pith/5XT3UJEFYDLNAMVNSO3TCMXG7G/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5XT3UJEFYDLNAMVNSO3TCMXG7G/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":"2026-08-11T01:49:53Z","links":{"resolver":"https://pith.science/pith/5XT3UJEFYDLNAMVNSO3TCMXG7G","bundle":"https://pith.science/pith/5XT3UJEFYDLNAMVNSO3TCMXG7G/bundle.json","state":"https://pith.science/pith/5XT3UJEFYDLNAMVNSO3TCMXG7G/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5XT3UJEFYDLNAMVNSO3TCMXG7G/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:5XT3UJEFYDLNAMVNSO3TCMXG7G","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":"db91ff6ca22a38dd945c6cc71c828a61e0e9ab5f2e1921faf3197bf0695c936a","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2026-07-28T03:40:20Z","title_canon_sha256":"c544e4da8c2b3dbea1fceea611e129a1aad13f92f666766dc9074eb17fd752bf"},"schema_version":"1.0","source":{"id":"2607.25246","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.25246","created_at":"2026-07-29T00:25:10Z"},{"alias_kind":"arxiv_version","alias_value":"2607.25246v1","created_at":"2026-07-29T00:25:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.25246","created_at":"2026-07-29T00:25:10Z"},{"alias_kind":"pith_short_12","alias_value":"5XT3UJEFYDLN","created_at":"2026-07-29T00:25:10Z"},{"alias_kind":"pith_short_16","alias_value":"5XT3UJEFYDLNAMVN","created_at":"2026-07-29T00:25:10Z"},{"alias_kind":"pith_short_8","alias_value":"5XT3UJEF","created_at":"2026-07-29T00:25:10Z"}],"graph_snapshots":[{"event_id":"sha256:f70d30be43e006c4caea874861ff82f93abc99bd230e6aeed27d74ca131c9e9f","target":"graph","created_at":"2026-07-29T00:25:10Z","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/2607.25246/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we study the Nijenhuis modules of Nijenhuis algebras. The concepts of free, projective, injective, and flat Nijenhuis modules are introduced, and a construction of free Nijenhuis modules is given. The ring of Nijenhuis operators is introduced, and the relationship between Nijenhuis modules and modules over this ring is established by proving an isomorphism of the corresponding categories. Finally, it is proved that the category of Nijenhuis modules has enough projective, injective, and flat objects.","authors_text":"Liuqing Dong, Shilong Zhang, Xiao-Song Peng","cross_cats":["math.RA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2026-07-28T03:40:20Z","title":"Nijenhuis modules and the ring of Nijenhuis operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.25246","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":"sha256:7e020efa55f62de9fba913d6b05c5af0c7ed67479c35cd8ac7da6158145cfdf5","target":"record","created_at":"2026-07-29T00:25:10Z","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":"db91ff6ca22a38dd945c6cc71c828a61e0e9ab5f2e1921faf3197bf0695c936a","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2026-07-28T03:40:20Z","title_canon_sha256":"c544e4da8c2b3dbea1fceea611e129a1aad13f92f666766dc9074eb17fd752bf"},"schema_version":"1.0","source":{"id":"2607.25246","kind":"arxiv","version":1}},"canonical_sha256":"ede7ba2485c0d6d032ad93b73132e6f986bf2bb1e9cda5641edb3ad50bc1bfa9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ede7ba2485c0d6d032ad93b73132e6f986bf2bb1e9cda5641edb3ad50bc1bfa9","first_computed_at":"2026-07-29T00:25:10.599113Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-29T00:25:10.599113Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lC/Bz8uNki9QFz94mh/Ur1nCEJi5tToRGg4KCYl6BbCrSFJJydI9uU3KE5jah2+ny/DPy3a1FixZi01WLcA+Dg==","signature_status":"signed_v1","signed_at":"2026-07-29T00:25:10.599971Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.25246","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7e020efa55f62de9fba913d6b05c5af0c7ed67479c35cd8ac7da6158145cfdf5","sha256:f70d30be43e006c4caea874861ff82f93abc99bd230e6aeed27d74ca131c9e9f"],"state_sha256":"c5ccade2eaa4bfe37d91936915f15b835fc20fe1462a6a18a8ec4db5ee7eeb85"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4Kf43zTUXj459S1AZ9uGZfeZSpZ2qfb3vhRcPQ2pb3uc7QZZYaepwd1YEAKyr+F9KEhuqV6TYQtrIhRU7P3MBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-11T01:49:53.432296Z","bundle_sha256":"2786bb0b317e996a67825a4f0cfd4e2e99528d68d99664752a7b32446d5dd98c"}}