{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:AICGJZIXXDFJ4SDBZH6EGQJFDV","short_pith_number":"pith:AICGJZIX","canonical_record":{"source":{"id":"1505.02819","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-05-11T22:07:04Z","cross_cats_sorted":["math-ph","math.CA","math.DG","math.MP","math.PR"],"title_canon_sha256":"8bddae0d64a68b25cf92c70828c105e1654adb29e70c602e81fda18f5ffc922c","abstract_canon_sha256":"a1463ba1e2aa541a84b98903ac84a346e583b896f1011caf994be4ecb550bdf9"},"schema_version":"1.0"},"canonical_sha256":"020464e517b8ca9e4861c9fc4341251d69aab04c122e299e0743d88dcd53f45d","source":{"kind":"arxiv","id":"1505.02819","version":5},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.02819","created_at":"2026-05-18T00:58:35Z"},{"alias_kind":"arxiv_version","alias_value":"1505.02819v5","created_at":"2026-05-18T00:58:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.02819","created_at":"2026-05-18T00:58:35Z"},{"alias_kind":"pith_short_12","alias_value":"AICGJZIXXDFJ","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_16","alias_value":"AICGJZIXXDFJ4SDB","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_8","alias_value":"AICGJZIX","created_at":"2026-05-18T12:29:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:AICGJZIXXDFJ4SDBZH6EGQJFDV","target":"record","payload":{"canonical_record":{"source":{"id":"1505.02819","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-05-11T22:07:04Z","cross_cats_sorted":["math-ph","math.CA","math.DG","math.MP","math.PR"],"title_canon_sha256":"8bddae0d64a68b25cf92c70828c105e1654adb29e70c602e81fda18f5ffc922c","abstract_canon_sha256":"a1463ba1e2aa541a84b98903ac84a346e583b896f1011caf994be4ecb550bdf9"},"schema_version":"1.0"},"canonical_sha256":"020464e517b8ca9e4861c9fc4341251d69aab04c122e299e0743d88dcd53f45d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:58:35.630256Z","signature_b64":"HDY6bzOjK1+I0pBGU9D3H5NSoRTAYZnXQAD1pEymWv29HpNuEaoixeLC8/j9Wwn10//3ZOWfWeFc3WdjFtpIAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"020464e517b8ca9e4861c9fc4341251d69aab04c122e299e0743d88dcd53f45d","last_reissued_at":"2026-05-18T00:58:35.629568Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:58:35.629568Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1505.02819","source_version":5,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:58:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9I5lxl/b+o8KMg1LDoB63BkenhnuVIpU+gNj5rh6gCQ05wjIQ5/k6JYJY0J8el/1V+g9PZmsawPdR9sQumsfCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T17:29:38.700444Z"},"content_sha256":"139266723e97bc06ef5c7296c8cdd11a2c082d56c699bb8b0eb47744fd31c71d","schema_version":"1.0","event_id":"sha256:139266723e97bc06ef5c7296c8cdd11a2c082d56c699bb8b0eb47744fd31c71d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:AICGJZIXXDFJ4SDBZH6EGQJFDV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Densely defined non-closable curl on carpet-like metric measure spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.CA","math.DG","math.MP","math.PR"],"primary_cat":"math.FA","authors_text":"Alexander Teplyaev, Michael Hinz","submitted_at":"2015-05-11T22:07:04Z","abstract_excerpt":"The paper deals with the possibly degenerate behaviour of the exterior derivative operator defined on $1$-forms on metric measure spaces. The main examples we consider are the non self-similar Sierpinski carpets recently introduced by Mackay, Tyson and Wildrick. Although topologically one-dimensional, they may have positive two-dimensional Lebesgue measure and carry nontrivial $2$-forms. We prove that in this case the curl operator (and therefore also the exterior derivative on $1$-forms) is not closable, and that its adjoint operator has a trivial domain. We also formulate a similar more abst"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.02819","kind":"arxiv","version":5},"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":"2026-05-18T00:58:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hf8dZ7T9oV+exMjBorGthum+YnHX9L74gsZA2LtS5GrCDYDVcTpv1z3ICHJp2INgj2J9Eixo5MJTG7eVD45XAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T17:29:38.701102Z"},"content_sha256":"5317fcbe08dd850eb3f966482383108095bc6fd1932dc68ebaf8e670e953ec9a","schema_version":"1.0","event_id":"sha256:5317fcbe08dd850eb3f966482383108095bc6fd1932dc68ebaf8e670e953ec9a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AICGJZIXXDFJ4SDBZH6EGQJFDV/bundle.json","state_url":"https://pith.science/pith/AICGJZIXXDFJ4SDBZH6EGQJFDV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AICGJZIXXDFJ4SDBZH6EGQJFDV/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":"2026-05-29T17:29:38Z","links":{"resolver":"https://pith.science/pith/AICGJZIXXDFJ4SDBZH6EGQJFDV","bundle":"https://pith.science/pith/AICGJZIXXDFJ4SDBZH6EGQJFDV/bundle.json","state":"https://pith.science/pith/AICGJZIXXDFJ4SDBZH6EGQJFDV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AICGJZIXXDFJ4SDBZH6EGQJFDV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:AICGJZIXXDFJ4SDBZH6EGQJFDV","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":"a1463ba1e2aa541a84b98903ac84a346e583b896f1011caf994be4ecb550bdf9","cross_cats_sorted":["math-ph","math.CA","math.DG","math.MP","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-05-11T22:07:04Z","title_canon_sha256":"8bddae0d64a68b25cf92c70828c105e1654adb29e70c602e81fda18f5ffc922c"},"schema_version":"1.0","source":{"id":"1505.02819","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.02819","created_at":"2026-05-18T00:58:35Z"},{"alias_kind":"arxiv_version","alias_value":"1505.02819v5","created_at":"2026-05-18T00:58:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.02819","created_at":"2026-05-18T00:58:35Z"},{"alias_kind":"pith_short_12","alias_value":"AICGJZIXXDFJ","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_16","alias_value":"AICGJZIXXDFJ4SDB","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_8","alias_value":"AICGJZIX","created_at":"2026-05-18T12:29:10Z"}],"graph_snapshots":[{"event_id":"sha256:5317fcbe08dd850eb3f966482383108095bc6fd1932dc68ebaf8e670e953ec9a","target":"graph","created_at":"2026-05-18T00:58:35Z","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":"The paper deals with the possibly degenerate behaviour of the exterior derivative operator defined on $1$-forms on metric measure spaces. The main examples we consider are the non self-similar Sierpinski carpets recently introduced by Mackay, Tyson and Wildrick. Although topologically one-dimensional, they may have positive two-dimensional Lebesgue measure and carry nontrivial $2$-forms. We prove that in this case the curl operator (and therefore also the exterior derivative on $1$-forms) is not closable, and that its adjoint operator has a trivial domain. We also formulate a similar more abst","authors_text":"Alexander Teplyaev, Michael Hinz","cross_cats":["math-ph","math.CA","math.DG","math.MP","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-05-11T22:07:04Z","title":"Densely defined non-closable curl on carpet-like metric measure spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.02819","kind":"arxiv","version":5},"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":"sha256:139266723e97bc06ef5c7296c8cdd11a2c082d56c699bb8b0eb47744fd31c71d","target":"record","created_at":"2026-05-18T00:58:35Z","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":"a1463ba1e2aa541a84b98903ac84a346e583b896f1011caf994be4ecb550bdf9","cross_cats_sorted":["math-ph","math.CA","math.DG","math.MP","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-05-11T22:07:04Z","title_canon_sha256":"8bddae0d64a68b25cf92c70828c105e1654adb29e70c602e81fda18f5ffc922c"},"schema_version":"1.0","source":{"id":"1505.02819","kind":"arxiv","version":5}},"canonical_sha256":"020464e517b8ca9e4861c9fc4341251d69aab04c122e299e0743d88dcd53f45d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"020464e517b8ca9e4861c9fc4341251d69aab04c122e299e0743d88dcd53f45d","first_computed_at":"2026-05-18T00:58:35.629568Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:58:35.629568Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HDY6bzOjK1+I0pBGU9D3H5NSoRTAYZnXQAD1pEymWv29HpNuEaoixeLC8/j9Wwn10//3ZOWfWeFc3WdjFtpIAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:58:35.630256Z","signed_message":"canonical_sha256_bytes"},"source_id":"1505.02819","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:139266723e97bc06ef5c7296c8cdd11a2c082d56c699bb8b0eb47744fd31c71d","sha256:5317fcbe08dd850eb3f966482383108095bc6fd1932dc68ebaf8e670e953ec9a"],"state_sha256":"c62cc6abc7ef2f0b751284d91a4c6171e8e77e9caea1f876d395013f314abee9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Pw06LZrTg5wt4ccRrK3U7b6ZTAneaYhf8Rz6IHC5EQVvUeLchTCJwVVL3zesYyTIaT5KrcvouH5lpiDWu/tTBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-29T17:29:38.704243Z","bundle_sha256":"0e50a4e2372c0061342fac620e65e15eae24e8c39fa18a712d08c939b0c62d5f"}}