{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:CZYZSUXIPSYKW2IWKX3762SNV7","short_pith_number":"pith:CZYZSUXI","canonical_record":{"source":{"id":"2007.00670","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2020-07-01T18:00:02Z","cross_cats_sorted":["cond-mat.str-el","hep-th","math.OA","math.QA"],"title_canon_sha256":"d4b8408c9badb457f1c0bdc8686abc52297900bbc6d82ff15f6be2b8b593f6c4","abstract_canon_sha256":"059da3ec0c32a10fec85c4d380c816251af6009a2a110b709fdecb2457f7d70e"},"schema_version":"1.0"},"canonical_sha256":"16719952e87cb0ab691655f7ff6a4dafd97e1868a3305e5facafc6d8c3b552a8","source":{"kind":"arxiv","id":"2007.00670","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2007.00670","created_at":"2026-07-05T03:20:32Z"},{"alias_kind":"arxiv_version","alias_value":"2007.00670v3","created_at":"2026-07-05T03:20:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2007.00670","created_at":"2026-07-05T03:20:32Z"},{"alias_kind":"pith_short_12","alias_value":"CZYZSUXIPSYK","created_at":"2026-07-05T03:20:32Z"},{"alias_kind":"pith_short_16","alias_value":"CZYZSUXIPSYKW2IW","created_at":"2026-07-05T03:20:32Z"},{"alias_kind":"pith_short_8","alias_value":"CZYZSUXI","created_at":"2026-07-05T03:20:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:CZYZSUXIPSYKW2IWKX3762SNV7","target":"record","payload":{"canonical_record":{"source":{"id":"2007.00670","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2020-07-01T18:00:02Z","cross_cats_sorted":["cond-mat.str-el","hep-th","math.OA","math.QA"],"title_canon_sha256":"d4b8408c9badb457f1c0bdc8686abc52297900bbc6d82ff15f6be2b8b593f6c4","abstract_canon_sha256":"059da3ec0c32a10fec85c4d380c816251af6009a2a110b709fdecb2457f7d70e"},"schema_version":"1.0"},"canonical_sha256":"16719952e87cb0ab691655f7ff6a4dafd97e1868a3305e5facafc6d8c3b552a8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:20:32.830460Z","signature_b64":"0nw3dIvd+4A7gA7BV9VacSsQciENxSzXRUujSCxFN73I5WR/WbKjE35pD95/XvcJlAj3GDY3YxzW3lQersxqBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"16719952e87cb0ab691655f7ff6a4dafd97e1868a3305e5facafc6d8c3b552a8","last_reissued_at":"2026-07-05T03:20:32.829998Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:20:32.829998Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2007.00670","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T03:20:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8mG5tbc4lND6KQxkwdnvT3efw3ijIBJ2tcrFzl0Vw70SPW1Hondw+7XsHzcQLJIKHT96EMLFg6JHifaJt/PsCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T10:34:09.884034Z"},"content_sha256":"b8b6967284ba43c04450d57237d31486ff8cf6de2d52a29a2425e6ca66286500","schema_version":"1.0","event_id":"sha256:b8b6967284ba43c04450d57237d31486ff8cf6de2d52a29a2425e6ca66286500"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:CZYZSUXIPSYKW2IWKX3762SNV7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The $F$-Symbols for Transparent Haagerup-Izumi Categories with $G = \\mathbb{Z}_{2n+1}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.str-el","hep-th","math.OA","math.QA"],"primary_cat":"math.CT","authors_text":"Tzu-Chen Huang, Ying-Hsuan Lin","submitted_at":"2020-07-01T18:00:02Z","abstract_excerpt":"A fusion category is called transparent if the associator involving any invertible object is the identity map. For the Haagerup-Izumi fusion rings with $G = \\mathbb{Z}_{2n+1}$ (the $\\mathbb{Z}_3$ case is the Haagerup fusion ring with six simple objects), the transparent ansatz reduces the number of independent $F$-symbols from order $\\mathcal{O}(n^6)$ to $\\mathcal{O}(n^2)$, rendering the pentagon identity practically solvable. Transparent Haagerup-Izumi fusion categories are thereby constructively classified up to $G = \\mathbb{Z}_9$, recovering all known Haagerup-Izumi fusion categories to thi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2007.00670","kind":"arxiv","version":3},"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/2007.00670/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":"2026-07-05T03:20:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lNx0Iua2ZCRKBz/tpIz8IVIFwqI0XeXVAJCBaOgms5Dq83XvITcDSwUQNLtPLJrPUWK13dkeetkyug5thQPLBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T10:34:09.884918Z"},"content_sha256":"f0142098c4bfe568488dce61a531405cf3a582a8f87ac76c0dd7672b46e4aca2","schema_version":"1.0","event_id":"sha256:f0142098c4bfe568488dce61a531405cf3a582a8f87ac76c0dd7672b46e4aca2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CZYZSUXIPSYKW2IWKX3762SNV7/bundle.json","state_url":"https://pith.science/pith/CZYZSUXIPSYKW2IWKX3762SNV7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CZYZSUXIPSYKW2IWKX3762SNV7/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":"2026-08-09T10:34:09Z","links":{"resolver":"https://pith.science/pith/CZYZSUXIPSYKW2IWKX3762SNV7","bundle":"https://pith.science/pith/CZYZSUXIPSYKW2IWKX3762SNV7/bundle.json","state":"https://pith.science/pith/CZYZSUXIPSYKW2IWKX3762SNV7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CZYZSUXIPSYKW2IWKX3762SNV7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:CZYZSUXIPSYKW2IWKX3762SNV7","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":"059da3ec0c32a10fec85c4d380c816251af6009a2a110b709fdecb2457f7d70e","cross_cats_sorted":["cond-mat.str-el","hep-th","math.OA","math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2020-07-01T18:00:02Z","title_canon_sha256":"d4b8408c9badb457f1c0bdc8686abc52297900bbc6d82ff15f6be2b8b593f6c4"},"schema_version":"1.0","source":{"id":"2007.00670","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2007.00670","created_at":"2026-07-05T03:20:32Z"},{"alias_kind":"arxiv_version","alias_value":"2007.00670v3","created_at":"2026-07-05T03:20:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2007.00670","created_at":"2026-07-05T03:20:32Z"},{"alias_kind":"pith_short_12","alias_value":"CZYZSUXIPSYK","created_at":"2026-07-05T03:20:32Z"},{"alias_kind":"pith_short_16","alias_value":"CZYZSUXIPSYKW2IW","created_at":"2026-07-05T03:20:32Z"},{"alias_kind":"pith_short_8","alias_value":"CZYZSUXI","created_at":"2026-07-05T03:20:32Z"}],"graph_snapshots":[{"event_id":"sha256:f0142098c4bfe568488dce61a531405cf3a582a8f87ac76c0dd7672b46e4aca2","target":"graph","created_at":"2026-07-05T03:20:32Z","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/2007.00670/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A fusion category is called transparent if the associator involving any invertible object is the identity map. For the Haagerup-Izumi fusion rings with $G = \\mathbb{Z}_{2n+1}$ (the $\\mathbb{Z}_3$ case is the Haagerup fusion ring with six simple objects), the transparent ansatz reduces the number of independent $F$-symbols from order $\\mathcal{O}(n^6)$ to $\\mathcal{O}(n^2)$, rendering the pentagon identity practically solvable. Transparent Haagerup-Izumi fusion categories are thereby constructively classified up to $G = \\mathbb{Z}_9$, recovering all known Haagerup-Izumi fusion categories to thi","authors_text":"Tzu-Chen Huang, Ying-Hsuan Lin","cross_cats":["cond-mat.str-el","hep-th","math.OA","math.QA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2020-07-01T18:00:02Z","title":"The $F$-Symbols for Transparent Haagerup-Izumi Categories with $G = \\mathbb{Z}_{2n+1}$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2007.00670","kind":"arxiv","version":3},"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:b8b6967284ba43c04450d57237d31486ff8cf6de2d52a29a2425e6ca66286500","target":"record","created_at":"2026-07-05T03:20:32Z","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":"059da3ec0c32a10fec85c4d380c816251af6009a2a110b709fdecb2457f7d70e","cross_cats_sorted":["cond-mat.str-el","hep-th","math.OA","math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2020-07-01T18:00:02Z","title_canon_sha256":"d4b8408c9badb457f1c0bdc8686abc52297900bbc6d82ff15f6be2b8b593f6c4"},"schema_version":"1.0","source":{"id":"2007.00670","kind":"arxiv","version":3}},"canonical_sha256":"16719952e87cb0ab691655f7ff6a4dafd97e1868a3305e5facafc6d8c3b552a8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"16719952e87cb0ab691655f7ff6a4dafd97e1868a3305e5facafc6d8c3b552a8","first_computed_at":"2026-07-05T03:20:32.829998Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:20:32.829998Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0nw3dIvd+4A7gA7BV9VacSsQciENxSzXRUujSCxFN73I5WR/WbKjE35pD95/XvcJlAj3GDY3YxzW3lQersxqBA==","signature_status":"signed_v1","signed_at":"2026-07-05T03:20:32.830460Z","signed_message":"canonical_sha256_bytes"},"source_id":"2007.00670","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b8b6967284ba43c04450d57237d31486ff8cf6de2d52a29a2425e6ca66286500","sha256:f0142098c4bfe568488dce61a531405cf3a582a8f87ac76c0dd7672b46e4aca2"],"state_sha256":"bca3147f8fa1dc0fde0c6613bd8bda3a9e6f93207d9a3f2a9bf3424ecbeebe91"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9RyMFz+rx7rl1DN5n8l7BW+MP60WgnBuaHNpGcJ+uKU1DL14zVqRyyYnw7YHMv0DvVlgjGEc0kITRJ4kIYEABA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T10:34:09.897192Z","bundle_sha256":"7ec0c5d12b988ec865ee36934eff5b9ff1dde1018776017edccb69a2b586c656"}}