{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:424ZHMETK4EEBVQCOL2V7J4DMF","short_pith_number":"pith:424ZHMET","canonical_record":{"source":{"id":"1207.6440","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-07-27T04:02:34Z","cross_cats_sorted":[],"title_canon_sha256":"87fe942e30767d036bc47d14abf2ae6eacba878c971f21de5f52ac748af00f75","abstract_canon_sha256":"5ed619d0cea3c4493018188a666afce12d616cc10941661246323cfdd83d184c"},"schema_version":"1.0"},"canonical_sha256":"e6b993b093570840d60272f55fa783615016a79bf4e1d8aa4d3594ca02760c2c","source":{"kind":"arxiv","id":"1207.6440","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.6440","created_at":"2026-05-18T01:19:31Z"},{"alias_kind":"arxiv_version","alias_value":"1207.6440v1","created_at":"2026-05-18T01:19:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.6440","created_at":"2026-05-18T01:19:31Z"},{"alias_kind":"pith_short_12","alias_value":"424ZHMETK4EE","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"424ZHMETK4EEBVQC","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"424ZHMET","created_at":"2026-05-18T12:26:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:424ZHMETK4EEBVQCOL2V7J4DMF","target":"record","payload":{"canonical_record":{"source":{"id":"1207.6440","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-07-27T04:02:34Z","cross_cats_sorted":[],"title_canon_sha256":"87fe942e30767d036bc47d14abf2ae6eacba878c971f21de5f52ac748af00f75","abstract_canon_sha256":"5ed619d0cea3c4493018188a666afce12d616cc10941661246323cfdd83d184c"},"schema_version":"1.0"},"canonical_sha256":"e6b993b093570840d60272f55fa783615016a79bf4e1d8aa4d3594ca02760c2c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:19:31.112030Z","signature_b64":"36JvhPCCedpG+tBtt/YoTGQIr9u9fqRA30lvfNPi82//E/2396mXc8fmszfwkrS4lyH3Vm+xuMNgf8rkSVMRDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e6b993b093570840d60272f55fa783615016a79bf4e1d8aa4d3594ca02760c2c","last_reissued_at":"2026-05-18T01:19:31.111301Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:19:31.111301Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1207.6440","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":"2026-05-18T01:19:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Gfj7PvrhdmA0lh5+pY0ylpzqjid4xU4p/2S/ASx5q4GKkAau6LabPDJ9tfVN24wXWa4VyiDKLAxO20ASVW/uAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T13:46:26.192143Z"},"content_sha256":"3d378fe87547a96e29d43f472294e3d24e537cc4eddfe82598cf94bb90352966","schema_version":"1.0","event_id":"sha256:3d378fe87547a96e29d43f472294e3d24e537cc4eddfe82598cf94bb90352966"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:424ZHMETK4EEBVQCOL2V7J4DMF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Density problems on vector bundles and manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Lashi Bandara","submitted_at":"2012-07-27T04:02:34Z","abstract_excerpt":"We study some canonical differential operators on vector bundles over smooth, complete Riemannian manifolds. Under very general assumptions, we show that smooth, compactly supported sections are dense in the domains of these operators. Furthermore, we show that smooth, compactly supported functions are dense in second order Sobolev spaces on such manifolds under the sole additional assumption that the Ricci curvature is uniformly bounded from below."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.6440","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":"2026-05-18T01:19:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0T8X500fIZ+4wiHcInpB8Dry584+oX6mrV4iYFuTQVwxcZvCURbX7zCV7Afbp7amSWWZVl8hOh9D2PaZo47kAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T13:46:26.192684Z"},"content_sha256":"c5129aa70a4545b0fef3d904bae2da14cf3459579f7acf6418c21d8403d58ad2","schema_version":"1.0","event_id":"sha256:c5129aa70a4545b0fef3d904bae2da14cf3459579f7acf6418c21d8403d58ad2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/424ZHMETK4EEBVQCOL2V7J4DMF/bundle.json","state_url":"https://pith.science/pith/424ZHMETK4EEBVQCOL2V7J4DMF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/424ZHMETK4EEBVQCOL2V7J4DMF/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":"2026-05-27T13:46:26Z","links":{"resolver":"https://pith.science/pith/424ZHMETK4EEBVQCOL2V7J4DMF","bundle":"https://pith.science/pith/424ZHMETK4EEBVQCOL2V7J4DMF/bundle.json","state":"https://pith.science/pith/424ZHMETK4EEBVQCOL2V7J4DMF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/424ZHMETK4EEBVQCOL2V7J4DMF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:424ZHMETK4EEBVQCOL2V7J4DMF","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":"5ed619d0cea3c4493018188a666afce12d616cc10941661246323cfdd83d184c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-07-27T04:02:34Z","title_canon_sha256":"87fe942e30767d036bc47d14abf2ae6eacba878c971f21de5f52ac748af00f75"},"schema_version":"1.0","source":{"id":"1207.6440","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.6440","created_at":"2026-05-18T01:19:31Z"},{"alias_kind":"arxiv_version","alias_value":"1207.6440v1","created_at":"2026-05-18T01:19:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.6440","created_at":"2026-05-18T01:19:31Z"},{"alias_kind":"pith_short_12","alias_value":"424ZHMETK4EE","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"424ZHMETK4EEBVQC","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"424ZHMET","created_at":"2026-05-18T12:26:53Z"}],"graph_snapshots":[{"event_id":"sha256:c5129aa70a4545b0fef3d904bae2da14cf3459579f7acf6418c21d8403d58ad2","target":"graph","created_at":"2026-05-18T01:19:31Z","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 study some canonical differential operators on vector bundles over smooth, complete Riemannian manifolds. Under very general assumptions, we show that smooth, compactly supported sections are dense in the domains of these operators. Furthermore, we show that smooth, compactly supported functions are dense in second order Sobolev spaces on such manifolds under the sole additional assumption that the Ricci curvature is uniformly bounded from below.","authors_text":"Lashi Bandara","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-07-27T04:02:34Z","title":"Density problems on vector bundles and manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.6440","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":"sha256:3d378fe87547a96e29d43f472294e3d24e537cc4eddfe82598cf94bb90352966","target":"record","created_at":"2026-05-18T01:19:31Z","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":"5ed619d0cea3c4493018188a666afce12d616cc10941661246323cfdd83d184c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-07-27T04:02:34Z","title_canon_sha256":"87fe942e30767d036bc47d14abf2ae6eacba878c971f21de5f52ac748af00f75"},"schema_version":"1.0","source":{"id":"1207.6440","kind":"arxiv","version":1}},"canonical_sha256":"e6b993b093570840d60272f55fa783615016a79bf4e1d8aa4d3594ca02760c2c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e6b993b093570840d60272f55fa783615016a79bf4e1d8aa4d3594ca02760c2c","first_computed_at":"2026-05-18T01:19:31.111301Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:19:31.111301Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"36JvhPCCedpG+tBtt/YoTGQIr9u9fqRA30lvfNPi82//E/2396mXc8fmszfwkrS4lyH3Vm+xuMNgf8rkSVMRDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:19:31.112030Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.6440","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3d378fe87547a96e29d43f472294e3d24e537cc4eddfe82598cf94bb90352966","sha256:c5129aa70a4545b0fef3d904bae2da14cf3459579f7acf6418c21d8403d58ad2"],"state_sha256":"cf18d14f20b70574caaef3c510071a5bfbb7c20b025afa1affd3272b68ec9cc3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rVGr4DPz879H0mwpsagRtVfFq+W1gsjFk/1jFpOOyM+4J5Wp4k4rS6Udas85k3aHRib4CpJISaAtWNWSNQkgAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T13:46:26.195511Z","bundle_sha256":"a9b003b041e5cebf6846833ee7114c919e9a71ddd11cce7380f215c7a8969361"}}