{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:A5Q2RHMROESJT4H6OHEWVC2ZEW","short_pith_number":"pith:A5Q2RHMR","canonical_record":{"source":{"id":"2507.06962","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2025-07-09T15:49:07Z","cross_cats_sorted":["math.CT","math.FA"],"title_canon_sha256":"0483999ee4e0c1aa3e703debab94f94acfd1f7c827e091bfc7cbddc31125ef86","abstract_canon_sha256":"854ea058f85a95ae567b80c829ac5839a26e382a72304875fd35b501546ceb8d"},"schema_version":"1.0"},"canonical_sha256":"0761a89d91712499f0fe71c96a8b59259bdc8c7d34299cdc5b86f7afd2148e70","source":{"kind":"arxiv","id":"2507.06962","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.06962","created_at":"2026-07-05T11:37:25Z"},{"alias_kind":"arxiv_version","alias_value":"2507.06962v2","created_at":"2026-07-05T11:37:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.06962","created_at":"2026-07-05T11:37:25Z"},{"alias_kind":"pith_short_12","alias_value":"A5Q2RHMROESJ","created_at":"2026-07-05T11:37:25Z"},{"alias_kind":"pith_short_16","alias_value":"A5Q2RHMROESJT4H6","created_at":"2026-07-05T11:37:25Z"},{"alias_kind":"pith_short_8","alias_value":"A5Q2RHMR","created_at":"2026-07-05T11:37:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:A5Q2RHMROESJT4H6OHEWVC2ZEW","target":"record","payload":{"canonical_record":{"source":{"id":"2507.06962","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2025-07-09T15:49:07Z","cross_cats_sorted":["math.CT","math.FA"],"title_canon_sha256":"0483999ee4e0c1aa3e703debab94f94acfd1f7c827e091bfc7cbddc31125ef86","abstract_canon_sha256":"854ea058f85a95ae567b80c829ac5839a26e382a72304875fd35b501546ceb8d"},"schema_version":"1.0"},"canonical_sha256":"0761a89d91712499f0fe71c96a8b59259bdc8c7d34299cdc5b86f7afd2148e70","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:37:25.511190Z","signature_b64":"mUWKA/x1kgTaVn1iHNVjPP7skXzqS54gxsuSHOoivrejIEIv8jn4vDsLW0ZzPCC8CgT/6GcaSLuKN1ctZOTDBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0761a89d91712499f0fe71c96a8b59259bdc8c7d34299cdc5b86f7afd2148e70","last_reissued_at":"2026-07-05T11:37:25.510621Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:37:25.510621Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2507.06962","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":"2026-07-05T11:37:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nbcGUusQwwz30Nsrjx5hKGpPrYqNT19Y0zk1heYDCltTvfE+PGKRHdt9O1hsEWNvUULULzEdCHuznkX1QztcAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T08:15:54.311689Z"},"content_sha256":"ef787309588e834b5610b5b95c46a42a7d7c6f63c206fc78f70f26c9fff6fc4c","schema_version":"1.0","event_id":"sha256:ef787309588e834b5610b5b95c46a42a7d7c6f63c206fc78f70f26c9fff6fc4c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:A5Q2RHMROESJT4H6OHEWVC2ZEW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Normed representations of weight quivers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT","math.FA"],"primary_cat":"math.RT","authors_text":"Yu-Zhe Liu","submitted_at":"2025-07-09T15:49:07Z","abstract_excerpt":"Let $A$ and $B$ be two tensor rings given by weight quivers. We introduce norms for tensor rings and $(A,B)$-bimodules, and define an important category $\\mathscr{A}^p_{\\varsigma}$ in this paper whose object is a triple $(N,v,\\delta)$ given by an $(A,B)$-bimodule $N$, a special element $v\\in V$ satisfying some special conditions, and a special $(A,B)$-homomorphism $\\delta: N^{\\oplus_p 2^{\\dim A}} \\to N$ and each morphism $(N,v,\\delta) \\to (N',v',\\delta')$ is given by an $(A,B)$-homomorphism $\\theta: N\\to N'$ such that $\\theta(v)=v'$ and $\\delta' \\theta^{\\oplus 2^{\\dim A}} = \\theta\\delta$ hold."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.06962","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/2507.06962/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":"2026-07-05T11:37:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uw7xfeE4LUD4lzmgn9JgzvajkaNTYct7ZYQpLoBrv6519qofqA2PXsEsYTT/8m/XjIJuksV0nMUAg4ExcWoqBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T08:15:54.312210Z"},"content_sha256":"68926193b910c7467bb7079d6f81de66d4c20784058a1375387b40b6611ebd5c","schema_version":"1.0","event_id":"sha256:68926193b910c7467bb7079d6f81de66d4c20784058a1375387b40b6611ebd5c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/A5Q2RHMROESJT4H6OHEWVC2ZEW/bundle.json","state_url":"https://pith.science/pith/A5Q2RHMROESJT4H6OHEWVC2ZEW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/A5Q2RHMROESJT4H6OHEWVC2ZEW/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":"2026-08-16T08:15:54Z","links":{"resolver":"https://pith.science/pith/A5Q2RHMROESJT4H6OHEWVC2ZEW","bundle":"https://pith.science/pith/A5Q2RHMROESJT4H6OHEWVC2ZEW/bundle.json","state":"https://pith.science/pith/A5Q2RHMROESJT4H6OHEWVC2ZEW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/A5Q2RHMROESJT4H6OHEWVC2ZEW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:A5Q2RHMROESJT4H6OHEWVC2ZEW","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":"854ea058f85a95ae567b80c829ac5839a26e382a72304875fd35b501546ceb8d","cross_cats_sorted":["math.CT","math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2025-07-09T15:49:07Z","title_canon_sha256":"0483999ee4e0c1aa3e703debab94f94acfd1f7c827e091bfc7cbddc31125ef86"},"schema_version":"1.0","source":{"id":"2507.06962","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.06962","created_at":"2026-07-05T11:37:25Z"},{"alias_kind":"arxiv_version","alias_value":"2507.06962v2","created_at":"2026-07-05T11:37:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.06962","created_at":"2026-07-05T11:37:25Z"},{"alias_kind":"pith_short_12","alias_value":"A5Q2RHMROESJ","created_at":"2026-07-05T11:37:25Z"},{"alias_kind":"pith_short_16","alias_value":"A5Q2RHMROESJT4H6","created_at":"2026-07-05T11:37:25Z"},{"alias_kind":"pith_short_8","alias_value":"A5Q2RHMR","created_at":"2026-07-05T11:37:25Z"}],"graph_snapshots":[{"event_id":"sha256:68926193b910c7467bb7079d6f81de66d4c20784058a1375387b40b6611ebd5c","target":"graph","created_at":"2026-07-05T11:37:25Z","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/2507.06962/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $A$ and $B$ be two tensor rings given by weight quivers. We introduce norms for tensor rings and $(A,B)$-bimodules, and define an important category $\\mathscr{A}^p_{\\varsigma}$ in this paper whose object is a triple $(N,v,\\delta)$ given by an $(A,B)$-bimodule $N$, a special element $v\\in V$ satisfying some special conditions, and a special $(A,B)$-homomorphism $\\delta: N^{\\oplus_p 2^{\\dim A}} \\to N$ and each morphism $(N,v,\\delta) \\to (N',v',\\delta')$ is given by an $(A,B)$-homomorphism $\\theta: N\\to N'$ such that $\\theta(v)=v'$ and $\\delta' \\theta^{\\oplus 2^{\\dim A}} = \\theta\\delta$ hold.","authors_text":"Yu-Zhe Liu","cross_cats":["math.CT","math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2025-07-09T15:49:07Z","title":"Normed representations of weight quivers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.06962","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:ef787309588e834b5610b5b95c46a42a7d7c6f63c206fc78f70f26c9fff6fc4c","target":"record","created_at":"2026-07-05T11:37:25Z","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":"854ea058f85a95ae567b80c829ac5839a26e382a72304875fd35b501546ceb8d","cross_cats_sorted":["math.CT","math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2025-07-09T15:49:07Z","title_canon_sha256":"0483999ee4e0c1aa3e703debab94f94acfd1f7c827e091bfc7cbddc31125ef86"},"schema_version":"1.0","source":{"id":"2507.06962","kind":"arxiv","version":2}},"canonical_sha256":"0761a89d91712499f0fe71c96a8b59259bdc8c7d34299cdc5b86f7afd2148e70","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0761a89d91712499f0fe71c96a8b59259bdc8c7d34299cdc5b86f7afd2148e70","first_computed_at":"2026-07-05T11:37:25.510621Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:37:25.510621Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mUWKA/x1kgTaVn1iHNVjPP7skXzqS54gxsuSHOoivrejIEIv8jn4vDsLW0ZzPCC8CgT/6GcaSLuKN1ctZOTDBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T11:37:25.511190Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.06962","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ef787309588e834b5610b5b95c46a42a7d7c6f63c206fc78f70f26c9fff6fc4c","sha256:68926193b910c7467bb7079d6f81de66d4c20784058a1375387b40b6611ebd5c"],"state_sha256":"204e737375f5b8a6c87dd1621e9e4fc8726944729e87cf9179c5db3fb70a4974"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UOLqB6lOQeK9+3kCIUHkeaeHhPbFVPKxN0DfGWnLkDy5++e1f/0osSLgrpCXMevt7F73mJf1B6ATPLegCofiAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T08:15:54.318251Z","bundle_sha256":"17df2da9e09e8efa0b143c9c49fded9780adedc0ce01e0d463e7785cfd0386cb"}}