{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:34J2G74MY5CHYFOTRDHUVCODW7","short_pith_number":"pith:34J2G74M","canonical_record":{"source":{"id":"2507.02249","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2025-07-03T02:45:43Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"c303f726ee19e3bfda2f1b54a1e19a0334a8190265fa2f32480ccf1a48394c54","abstract_canon_sha256":"178fa7d0839198d0ac4f304498cd9c26948ecb65e28411fd9299f9503ba5b4ce"},"schema_version":"1.0"},"canonical_sha256":"df13a37f8cc7447c15d388cf4a89c3b7c2fcd6f8a13bde50fb6675fc5fe8bb76","source":{"kind":"arxiv","id":"2507.02249","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.02249","created_at":"2026-07-05T11:31:14Z"},{"alias_kind":"arxiv_version","alias_value":"2507.02249v1","created_at":"2026-07-05T11:31:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.02249","created_at":"2026-07-05T11:31:14Z"},{"alias_kind":"pith_short_12","alias_value":"34J2G74MY5CH","created_at":"2026-07-05T11:31:14Z"},{"alias_kind":"pith_short_16","alias_value":"34J2G74MY5CHYFOT","created_at":"2026-07-05T11:31:14Z"},{"alias_kind":"pith_short_8","alias_value":"34J2G74M","created_at":"2026-07-05T11:31:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:34J2G74MY5CHYFOTRDHUVCODW7","target":"record","payload":{"canonical_record":{"source":{"id":"2507.02249","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2025-07-03T02:45:43Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"c303f726ee19e3bfda2f1b54a1e19a0334a8190265fa2f32480ccf1a48394c54","abstract_canon_sha256":"178fa7d0839198d0ac4f304498cd9c26948ecb65e28411fd9299f9503ba5b4ce"},"schema_version":"1.0"},"canonical_sha256":"df13a37f8cc7447c15d388cf4a89c3b7c2fcd6f8a13bde50fb6675fc5fe8bb76","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:31:14.459370Z","signature_b64":"Wwy6X64vksDQ2ucy+t/mEU2EfFGC2beyYUv3fbDNxVhGyokTMyGKquPKS4g/rbAzFQvfyh5xMKtj65ubx/xPBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"df13a37f8cc7447c15d388cf4a89c3b7c2fcd6f8a13bde50fb6675fc5fe8bb76","last_reissued_at":"2026-07-05T11:31:14.458922Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:31:14.458922Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2507.02249","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":"2026-07-05T11:31:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iX8UioAP/VGojNctYsOPRuWwMmH4O1jPkKXA9C9vhtQ7mYhTyrdRuRtmEBFxVAbHNi1DVW1peeFLGHLJl4WDAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T14:11:38.117476Z"},"content_sha256":"b5044c8ae9ab44f55a7a534dbb8a3e0804065c365342751e38604bed970bc8f1","schema_version":"1.0","event_id":"sha256:b5044c8ae9ab44f55a7a534dbb8a3e0804065c365342751e38604bed970bc8f1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:34J2G74MY5CHYFOTRDHUVCODW7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Quasi-triangular and factorizable dendriform D-bialgebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.RA","authors_text":"You Wang","submitted_at":"2025-07-03T02:45:43Z","abstract_excerpt":"In this paper, we introduce the notions of quasi-triangular and factorizable dendriform D-bialgebras. A factorizable dendriform D-bialgebra leads to a factorization of the underlying dendriform algebra. We show that the dendriform double of a dendriform D-bialgebra naturally enjoys a factorizable dendriform D-bialgebra structure. Moreover, we introduce the notion of relative Rota-Baxter operators of nonzero weights on dendriform algebras and find that every quasi-triangular dendriform D-bialgebra can give rise to a relative Rota-Baxter operator of weight 1. Then we introduce the notion of quad"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.02249","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/2507.02249/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":"2026-07-05T11:31:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"O+5FRtrzwSJ0n5eLmcKkBsWCwn/WKIq0FYETVqIL8L2U6OpKxx4iBSSRG7sRiORwDQyFkPqbIYZp3FVZfMuiCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T14:11:38.118322Z"},"content_sha256":"4e873be18b68273e8316bc0a19d09eb7f4a1fe83a0877c21a5314ccf35059182","schema_version":"1.0","event_id":"sha256:4e873be18b68273e8316bc0a19d09eb7f4a1fe83a0877c21a5314ccf35059182"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/34J2G74MY5CHYFOTRDHUVCODW7/bundle.json","state_url":"https://pith.science/pith/34J2G74MY5CHYFOTRDHUVCODW7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/34J2G74MY5CHYFOTRDHUVCODW7/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":"2026-08-07T14:11:38Z","links":{"resolver":"https://pith.science/pith/34J2G74MY5CHYFOTRDHUVCODW7","bundle":"https://pith.science/pith/34J2G74MY5CHYFOTRDHUVCODW7/bundle.json","state":"https://pith.science/pith/34J2G74MY5CHYFOTRDHUVCODW7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/34J2G74MY5CHYFOTRDHUVCODW7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:34J2G74MY5CHYFOTRDHUVCODW7","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":"178fa7d0839198d0ac4f304498cd9c26948ecb65e28411fd9299f9503ba5b4ce","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2025-07-03T02:45:43Z","title_canon_sha256":"c303f726ee19e3bfda2f1b54a1e19a0334a8190265fa2f32480ccf1a48394c54"},"schema_version":"1.0","source":{"id":"2507.02249","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.02249","created_at":"2026-07-05T11:31:14Z"},{"alias_kind":"arxiv_version","alias_value":"2507.02249v1","created_at":"2026-07-05T11:31:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.02249","created_at":"2026-07-05T11:31:14Z"},{"alias_kind":"pith_short_12","alias_value":"34J2G74MY5CH","created_at":"2026-07-05T11:31:14Z"},{"alias_kind":"pith_short_16","alias_value":"34J2G74MY5CHYFOT","created_at":"2026-07-05T11:31:14Z"},{"alias_kind":"pith_short_8","alias_value":"34J2G74M","created_at":"2026-07-05T11:31:14Z"}],"graph_snapshots":[{"event_id":"sha256:4e873be18b68273e8316bc0a19d09eb7f4a1fe83a0877c21a5314ccf35059182","target":"graph","created_at":"2026-07-05T11:31:14Z","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/2507.02249/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we introduce the notions of quasi-triangular and factorizable dendriform D-bialgebras. A factorizable dendriform D-bialgebra leads to a factorization of the underlying dendriform algebra. We show that the dendriform double of a dendriform D-bialgebra naturally enjoys a factorizable dendriform D-bialgebra structure. Moreover, we introduce the notion of relative Rota-Baxter operators of nonzero weights on dendriform algebras and find that every quasi-triangular dendriform D-bialgebra can give rise to a relative Rota-Baxter operator of weight 1. Then we introduce the notion of quad","authors_text":"You Wang","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2025-07-03T02:45:43Z","title":"Quasi-triangular and factorizable dendriform D-bialgebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.02249","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:b5044c8ae9ab44f55a7a534dbb8a3e0804065c365342751e38604bed970bc8f1","target":"record","created_at":"2026-07-05T11:31:14Z","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":"178fa7d0839198d0ac4f304498cd9c26948ecb65e28411fd9299f9503ba5b4ce","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2025-07-03T02:45:43Z","title_canon_sha256":"c303f726ee19e3bfda2f1b54a1e19a0334a8190265fa2f32480ccf1a48394c54"},"schema_version":"1.0","source":{"id":"2507.02249","kind":"arxiv","version":1}},"canonical_sha256":"df13a37f8cc7447c15d388cf4a89c3b7c2fcd6f8a13bde50fb6675fc5fe8bb76","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"df13a37f8cc7447c15d388cf4a89c3b7c2fcd6f8a13bde50fb6675fc5fe8bb76","first_computed_at":"2026-07-05T11:31:14.458922Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:31:14.458922Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Wwy6X64vksDQ2ucy+t/mEU2EfFGC2beyYUv3fbDNxVhGyokTMyGKquPKS4g/rbAzFQvfyh5xMKtj65ubx/xPBg==","signature_status":"signed_v1","signed_at":"2026-07-05T11:31:14.459370Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.02249","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b5044c8ae9ab44f55a7a534dbb8a3e0804065c365342751e38604bed970bc8f1","sha256:4e873be18b68273e8316bc0a19d09eb7f4a1fe83a0877c21a5314ccf35059182"],"state_sha256":"3b29b2534d3a6b015c857082b95060f92558d44ee786de3d438495811dbb9f40"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8/BCZ7w7BqxbOrmisrUkNvRmesIo9zWJCzQiNH/ysDA9iKoXyzNKKS7QjY0/6M83D4HXgRTgcqCV6K/zArLYCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-07T14:11:38.123505Z","bundle_sha256":"8255a8ed6481e1190a7e7c244faa81d97e7353c5f5d4a725971feb942f5b8b38"}}