{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:SVCW365ZWJZSMNQUZU2FIYWGLD","short_pith_number":"pith:SVCW365Z","canonical_record":{"source":{"id":"1109.0894","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-09-05T13:29:37Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"670b5aeae803f0e16868ee70e22766031724b8a3db6f31835036e03ce8ccb6c7","abstract_canon_sha256":"60cffb3c04b1d44f7af964c81827828b799c6d6b68052f796b94147ae0c325fe"},"schema_version":"1.0"},"canonical_sha256":"95456dfbb9b273263614cd345462c658ef7fbea5c72c841bced35a78d484073f","source":{"kind":"arxiv","id":"1109.0894","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.0894","created_at":"2026-05-18T04:14:06Z"},{"alias_kind":"arxiv_version","alias_value":"1109.0894v1","created_at":"2026-05-18T04:14:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.0894","created_at":"2026-05-18T04:14:06Z"},{"alias_kind":"pith_short_12","alias_value":"SVCW365ZWJZS","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_16","alias_value":"SVCW365ZWJZSMNQU","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_8","alias_value":"SVCW365Z","created_at":"2026-05-18T12:26:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:SVCW365ZWJZSMNQUZU2FIYWGLD","target":"record","payload":{"canonical_record":{"source":{"id":"1109.0894","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-09-05T13:29:37Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"670b5aeae803f0e16868ee70e22766031724b8a3db6f31835036e03ce8ccb6c7","abstract_canon_sha256":"60cffb3c04b1d44f7af964c81827828b799c6d6b68052f796b94147ae0c325fe"},"schema_version":"1.0"},"canonical_sha256":"95456dfbb9b273263614cd345462c658ef7fbea5c72c841bced35a78d484073f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:14:06.428969Z","signature_b64":"vPM5Tip4laf1Atsg/ew9h42IyfFnBRCezS/L4BK6WFOpNFAbYsVATy0u3v1jVDEVAjyAQRrzcjKpcxAQZmk9BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"95456dfbb9b273263614cd345462c658ef7fbea5c72c841bced35a78d484073f","last_reissued_at":"2026-05-18T04:14:06.428399Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:14:06.428399Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1109.0894","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":"2026-05-18T04:14:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hasynk/MDkHmYjDXZK9wXOZp+yyEt25f0/HrVRCh2NuaL80bILkCFCXoA0e+Ow3o+gvbK+odvql6hSjkRvpmDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T19:14:15.055935Z"},"content_sha256":"95e9fdcc6f40776d6cbc3ecc3a23f8d2ad0733982cf5ac49621888495d297e82","schema_version":"1.0","event_id":"sha256:95e9fdcc6f40776d6cbc3ecc3a23f8d2ad0733982cf5ac49621888495d297e82"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:SVCW365ZWJZSMNQUZU2FIYWGLD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Generalized duality for k-forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.DG","authors_text":"Frank Klinker","submitted_at":"2011-09-05T13:29:37Z","abstract_excerpt":"We give the definition of a duality that is applicable to arbitrary $k$-forms. The operator that defines the duality depends on a fixed form $\\Omega$. Our definition extends in a very natural way the Hodge duality of $n$-forms in $2n$ dimensional spaces and the generalized duality of two-forms. We discuss the properties of the duality in the case where $\\Omega$ is invariant with respect to a subalgebra of $\\mathfrak{so}(V)$. Furthermore, we give examples for the invariant case as well as for the case of discrete symmetry."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.0894","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":""},"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":"2026-05-18T04:14:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c/KxoP0xL8gBVuQ3ayiiDoJsrbxR+xnJHJlf719BxffccFkSsMe4ZsBoZ473cHNp4sZk3i32jfz8WmbaSbtuCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T19:14:15.056282Z"},"content_sha256":"c3e6403c142b0d6e39535ac45d0acd20ed5ed7f724bc8501e4b8250874cfa01a","schema_version":"1.0","event_id":"sha256:c3e6403c142b0d6e39535ac45d0acd20ed5ed7f724bc8501e4b8250874cfa01a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SVCW365ZWJZSMNQUZU2FIYWGLD/bundle.json","state_url":"https://pith.science/pith/SVCW365ZWJZSMNQUZU2FIYWGLD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SVCW365ZWJZSMNQUZU2FIYWGLD/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":"2026-06-30T19:14:15Z","links":{"resolver":"https://pith.science/pith/SVCW365ZWJZSMNQUZU2FIYWGLD","bundle":"https://pith.science/pith/SVCW365ZWJZSMNQUZU2FIYWGLD/bundle.json","state":"https://pith.science/pith/SVCW365ZWJZSMNQUZU2FIYWGLD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SVCW365ZWJZSMNQUZU2FIYWGLD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:SVCW365ZWJZSMNQUZU2FIYWGLD","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":"60cffb3c04b1d44f7af964c81827828b799c6d6b68052f796b94147ae0c325fe","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-09-05T13:29:37Z","title_canon_sha256":"670b5aeae803f0e16868ee70e22766031724b8a3db6f31835036e03ce8ccb6c7"},"schema_version":"1.0","source":{"id":"1109.0894","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.0894","created_at":"2026-05-18T04:14:06Z"},{"alias_kind":"arxiv_version","alias_value":"1109.0894v1","created_at":"2026-05-18T04:14:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.0894","created_at":"2026-05-18T04:14:06Z"},{"alias_kind":"pith_short_12","alias_value":"SVCW365ZWJZS","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_16","alias_value":"SVCW365ZWJZSMNQU","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_8","alias_value":"SVCW365Z","created_at":"2026-05-18T12:26:41Z"}],"graph_snapshots":[{"event_id":"sha256:c3e6403c142b0d6e39535ac45d0acd20ed5ed7f724bc8501e4b8250874cfa01a","target":"graph","created_at":"2026-05-18T04:14:06Z","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"},"paper":{"abstract_excerpt":"We give the definition of a duality that is applicable to arbitrary $k$-forms. The operator that defines the duality depends on a fixed form $\\Omega$. Our definition extends in a very natural way the Hodge duality of $n$-forms in $2n$ dimensional spaces and the generalized duality of two-forms. We discuss the properties of the duality in the case where $\\Omega$ is invariant with respect to a subalgebra of $\\mathfrak{so}(V)$. Furthermore, we give examples for the invariant case as well as for the case of discrete symmetry.","authors_text":"Frank Klinker","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-09-05T13:29:37Z","title":"Generalized duality for k-forms"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.0894","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":"sha256:95e9fdcc6f40776d6cbc3ecc3a23f8d2ad0733982cf5ac49621888495d297e82","target":"record","created_at":"2026-05-18T04:14:06Z","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":"60cffb3c04b1d44f7af964c81827828b799c6d6b68052f796b94147ae0c325fe","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-09-05T13:29:37Z","title_canon_sha256":"670b5aeae803f0e16868ee70e22766031724b8a3db6f31835036e03ce8ccb6c7"},"schema_version":"1.0","source":{"id":"1109.0894","kind":"arxiv","version":1}},"canonical_sha256":"95456dfbb9b273263614cd345462c658ef7fbea5c72c841bced35a78d484073f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"95456dfbb9b273263614cd345462c658ef7fbea5c72c841bced35a78d484073f","first_computed_at":"2026-05-18T04:14:06.428399Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:14:06.428399Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vPM5Tip4laf1Atsg/ew9h42IyfFnBRCezS/L4BK6WFOpNFAbYsVATy0u3v1jVDEVAjyAQRrzcjKpcxAQZmk9BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:14:06.428969Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.0894","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:95e9fdcc6f40776d6cbc3ecc3a23f8d2ad0733982cf5ac49621888495d297e82","sha256:c3e6403c142b0d6e39535ac45d0acd20ed5ed7f724bc8501e4b8250874cfa01a"],"state_sha256":"a56a8b7caa9fc73adc75a00e97d1c7c6bbccbb4783af450167c90d919b04a286"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xGoWIEtuQ/A4YsYTRTaXlK1+tUe/pLLwCU0jxDXKhAfUx6ggjA4NI3nAKXwJGq6BlUybi7v4BYQXzcUqpjouCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-30T19:14:15.058390Z","bundle_sha256":"0e87bb068657c94b2baddc35dd9055068bb9f651174f84d4afd779761be6ef00"}}