{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:5URBJVI5BNXE6X2OCM3C4F3S2U","short_pith_number":"pith:5URBJVI5","canonical_record":{"source":{"id":"2112.02199","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2021-12-04T00:08:14Z","cross_cats_sorted":["hep-th","math.AT","math.MP"],"title_canon_sha256":"2da20b5329118a3a3b771f2f001b259ef4bb4db713c5d358ad4e0531c928f2a5","abstract_canon_sha256":"5d63a9ffdb2756ece8d21fbbbf178953ac6005843775e8772604fb7a8cc6a251"},"schema_version":"1.0"},"canonical_sha256":"ed2214d51d0b6e4f5f4e13362e1772d50559d9d50bfb5cb9c356dc5e72bfb333","source":{"kind":"arxiv","id":"2112.02199","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2112.02199","created_at":"2026-07-05T05:20:33Z"},{"alias_kind":"arxiv_version","alias_value":"2112.02199v1","created_at":"2026-07-05T05:20:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2112.02199","created_at":"2026-07-05T05:20:33Z"},{"alias_kind":"pith_short_12","alias_value":"5URBJVI5BNXE","created_at":"2026-07-05T05:20:33Z"},{"alias_kind":"pith_short_16","alias_value":"5URBJVI5BNXE6X2O","created_at":"2026-07-05T05:20:33Z"},{"alias_kind":"pith_short_8","alias_value":"5URBJVI5","created_at":"2026-07-05T05:20:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:5URBJVI5BNXE6X2OCM3C4F3S2U","target":"record","payload":{"canonical_record":{"source":{"id":"2112.02199","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2021-12-04T00:08:14Z","cross_cats_sorted":["hep-th","math.AT","math.MP"],"title_canon_sha256":"2da20b5329118a3a3b771f2f001b259ef4bb4db713c5d358ad4e0531c928f2a5","abstract_canon_sha256":"5d63a9ffdb2756ece8d21fbbbf178953ac6005843775e8772604fb7a8cc6a251"},"schema_version":"1.0"},"canonical_sha256":"ed2214d51d0b6e4f5f4e13362e1772d50559d9d50bfb5cb9c356dc5e72bfb333","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:20:33.529743Z","signature_b64":"EqAUS1E1gPyKYKas21CWIWTbL5gbhR9nVLaBuBKfAUg9de0IFwtEVb7BPdqGKnF/SgLA3cBSr5asXWaORLY7Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ed2214d51d0b6e4f5f4e13362e1772d50559d9d50bfb5cb9c356dc5e72bfb333","last_reissued_at":"2026-07-05T05:20:33.529336Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:20:33.529336Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2112.02199","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":"2026-07-05T05:20:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Il4A+636jdAg02NN9C84FcA7tlUL9zICvkVAGFfiLV9iBaKxEqC9XPugMqPwi2E0BUiLxBsaIPH7ZuROYU/pAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T14:11:53.859698Z"},"content_sha256":"5cb15a053a25946723124096c8c02b89f0862097e792b963b16ca79f92fa008a","schema_version":"1.0","event_id":"sha256:5cb15a053a25946723124096c8c02b89f0862097e792b963b16ca79f92fa008a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:5URBJVI5BNXE6X2OCM3C4F3S2U","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Abelian duality in topological field theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th","math.AT","math.MP"],"primary_cat":"math-ph","authors_text":"Yu Leon Liu","submitted_at":"2021-12-04T00:08:14Z","abstract_excerpt":"We prove a topological version of abelian duality where the gauge groups are finite abelian. The theories are finite homotopy TFTs, topological analogues of the $p$-form $U(1)$ gauge theories. Using Brown-Comenetz duality, we extend the duality to finite homotopy TFTs of $\\pi$-finite spectra."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2112.02199","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/2112.02199/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":"2026-07-05T05:20:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rx8dfDYaSCpYkFuKUjY8I3HA4kacLhuALa0qLBjF9+oHf8QFg+mVjkz75301xjD07UGywayd15BZLH8A08RaDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T14:11:53.860430Z"},"content_sha256":"f99d42ee7e5b7279e0ce009851b63163256602d28ad5e6b275f7da25a9b9f79b","schema_version":"1.0","event_id":"sha256:f99d42ee7e5b7279e0ce009851b63163256602d28ad5e6b275f7da25a9b9f79b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5URBJVI5BNXE6X2OCM3C4F3S2U/bundle.json","state_url":"https://pith.science/pith/5URBJVI5BNXE6X2OCM3C4F3S2U/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5URBJVI5BNXE6X2OCM3C4F3S2U/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":"2026-08-14T14:11:53Z","links":{"resolver":"https://pith.science/pith/5URBJVI5BNXE6X2OCM3C4F3S2U","bundle":"https://pith.science/pith/5URBJVI5BNXE6X2OCM3C4F3S2U/bundle.json","state":"https://pith.science/pith/5URBJVI5BNXE6X2OCM3C4F3S2U/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5URBJVI5BNXE6X2OCM3C4F3S2U/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:5URBJVI5BNXE6X2OCM3C4F3S2U","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":"5d63a9ffdb2756ece8d21fbbbf178953ac6005843775e8772604fb7a8cc6a251","cross_cats_sorted":["hep-th","math.AT","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2021-12-04T00:08:14Z","title_canon_sha256":"2da20b5329118a3a3b771f2f001b259ef4bb4db713c5d358ad4e0531c928f2a5"},"schema_version":"1.0","source":{"id":"2112.02199","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2112.02199","created_at":"2026-07-05T05:20:33Z"},{"alias_kind":"arxiv_version","alias_value":"2112.02199v1","created_at":"2026-07-05T05:20:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2112.02199","created_at":"2026-07-05T05:20:33Z"},{"alias_kind":"pith_short_12","alias_value":"5URBJVI5BNXE","created_at":"2026-07-05T05:20:33Z"},{"alias_kind":"pith_short_16","alias_value":"5URBJVI5BNXE6X2O","created_at":"2026-07-05T05:20:33Z"},{"alias_kind":"pith_short_8","alias_value":"5URBJVI5","created_at":"2026-07-05T05:20:33Z"}],"graph_snapshots":[{"event_id":"sha256:f99d42ee7e5b7279e0ce009851b63163256602d28ad5e6b275f7da25a9b9f79b","target":"graph","created_at":"2026-07-05T05:20:33Z","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/2112.02199/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove a topological version of abelian duality where the gauge groups are finite abelian. The theories are finite homotopy TFTs, topological analogues of the $p$-form $U(1)$ gauge theories. Using Brown-Comenetz duality, we extend the duality to finite homotopy TFTs of $\\pi$-finite spectra.","authors_text":"Yu Leon Liu","cross_cats":["hep-th","math.AT","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2021-12-04T00:08:14Z","title":"Abelian duality in topological field theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2112.02199","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":"sha256:5cb15a053a25946723124096c8c02b89f0862097e792b963b16ca79f92fa008a","target":"record","created_at":"2026-07-05T05:20:33Z","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":"5d63a9ffdb2756ece8d21fbbbf178953ac6005843775e8772604fb7a8cc6a251","cross_cats_sorted":["hep-th","math.AT","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2021-12-04T00:08:14Z","title_canon_sha256":"2da20b5329118a3a3b771f2f001b259ef4bb4db713c5d358ad4e0531c928f2a5"},"schema_version":"1.0","source":{"id":"2112.02199","kind":"arxiv","version":1}},"canonical_sha256":"ed2214d51d0b6e4f5f4e13362e1772d50559d9d50bfb5cb9c356dc5e72bfb333","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ed2214d51d0b6e4f5f4e13362e1772d50559d9d50bfb5cb9c356dc5e72bfb333","first_computed_at":"2026-07-05T05:20:33.529336Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:20:33.529336Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EqAUS1E1gPyKYKas21CWIWTbL5gbhR9nVLaBuBKfAUg9de0IFwtEVb7BPdqGKnF/SgLA3cBSr5asXWaORLY7Dg==","signature_status":"signed_v1","signed_at":"2026-07-05T05:20:33.529743Z","signed_message":"canonical_sha256_bytes"},"source_id":"2112.02199","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5cb15a053a25946723124096c8c02b89f0862097e792b963b16ca79f92fa008a","sha256:f99d42ee7e5b7279e0ce009851b63163256602d28ad5e6b275f7da25a9b9f79b"],"state_sha256":"3834da02b244f90d061949450bce7c881c3c412c005435adeacab74957681828"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/xIA+k3SptbB0jWnAFm1bEbJrpfKHSVZGs+77YUEnKwCphBjgYdrsdMdzF6GYLBjhAyUjV9aojnVQkXkdT0TCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-14T14:11:53.868804Z","bundle_sha256":"04dc532f2ded7113f889e69fde0ed79a11042f342fb21f3cfe785a707fe63825"}}