{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:6WPLBLQG5UG2W4VEPZW5KRNPZU","short_pith_number":"pith:6WPLBLQG","canonical_record":{"source":{"id":"1009.5065","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-09-26T05:56:06Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"e808c5f56275af189a0ccf12c7b825e8b1a099087157bb30965b526f3ae25282","abstract_canon_sha256":"7036f31f944add93dae1cd6a9bbf0d9cea6279a02bc71acec3d894aaf1be68af"},"schema_version":"1.0"},"canonical_sha256":"f59eb0ae06ed0dab72a47e6dd545afcd27b6276325d2aa3c5e20cf85fcb8cba3","source":{"kind":"arxiv","id":"1009.5065","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.5065","created_at":"2026-05-18T03:03:33Z"},{"alias_kind":"arxiv_version","alias_value":"1009.5065v1","created_at":"2026-05-18T03:03:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.5065","created_at":"2026-05-18T03:03:33Z"},{"alias_kind":"pith_short_12","alias_value":"6WPLBLQG5UG2","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"6WPLBLQG5UG2W4VE","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"6WPLBLQG","created_at":"2026-05-18T12:26:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:6WPLBLQG5UG2W4VEPZW5KRNPZU","target":"record","payload":{"canonical_record":{"source":{"id":"1009.5065","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-09-26T05:56:06Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"e808c5f56275af189a0ccf12c7b825e8b1a099087157bb30965b526f3ae25282","abstract_canon_sha256":"7036f31f944add93dae1cd6a9bbf0d9cea6279a02bc71acec3d894aaf1be68af"},"schema_version":"1.0"},"canonical_sha256":"f59eb0ae06ed0dab72a47e6dd545afcd27b6276325d2aa3c5e20cf85fcb8cba3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:03:33.046953Z","signature_b64":"jwdq4CJhrcpNk7CEn8yW1PvTlEnQliDPjbEp2hPiOltz6QHpBptNTNO98simoGrWkOWVy55KFnxD3J4B5U23BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f59eb0ae06ed0dab72a47e6dd545afcd27b6276325d2aa3c5e20cf85fcb8cba3","last_reissued_at":"2026-05-18T03:03:33.046430Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:03:33.046430Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1009.5065","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":"2026-05-18T03:03:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bLQFH7fDOn0yRgQnkxAxl6l0VIpw1s9YWvgN9wtwx8s6Z+MbFz255yoCshu1oMHrzFHERnh0DWVqkE8WfTD6Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T07:47:26.791804Z"},"content_sha256":"81ddc133a3a0ce3ab7ec4ad8a9e96f465200abc617e28f1e2c31794c8260ae7c","schema_version":"1.0","event_id":"sha256:81ddc133a3a0ce3ab7ec4ad8a9e96f465200abc617e28f1e2c31794c8260ae7c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:6WPLBLQG5UG2W4VEPZW5KRNPZU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"$L_1$-uniqueness of degenerate elliptic operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Adam Sikora, Derek W. Robinson","submitted_at":"2010-09-26T05:56:06Z","abstract_excerpt":"Let $\\Omega$ be an open subset of $\\Ri^d$ with $0\\in \\Omega$.\n  Further let $H_\\Omega=-\\sum^d_{i,j=1}\\partial_i\\,c_{ij}\\,\\partial_j$ be a second-order partial differential operator with domain $C_c^\\infty(\\Omega)$ where the coefficients $c_{ij}\\in W^{1,\\infty}_{\\rm loc}(\\bar\\Omega)$ are real, $c_{ij}=c_{ji}$ and the coefficient matrix $C=(c_{ij})$ satisfies bounds $0<C(x)\\leq c(|x|) I$ for all $x\\in \\Omega$. If \\[ \\int^\\infty_0ds\\,s^{d/2}\\,e^{-\\lambda\\,\\mu(s)^2}<\\infty \\] for some $\\lambda>0$ where $\\mu(s)=\\int^s_0dt\\,c(t)^{-1/2}$ then we establish that $H_\\Omega$ is $L_1$-unique, i.e.\\ it has"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.5065","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":"2026-05-18T03:03:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Es68Z53Kxqsic2AKvJX0FLa4z3JhLBK7mz95Rj+hx2oXRTeslU0GNbFG5YAJAYc+n+4Od9zb2Tg+P4HH6Vy0DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T07:47:26.792146Z"},"content_sha256":"7a051fad03f0560dd55a077f0f8731536236bc9ceadb13fc101710e5ba3088b2","schema_version":"1.0","event_id":"sha256:7a051fad03f0560dd55a077f0f8731536236bc9ceadb13fc101710e5ba3088b2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6WPLBLQG5UG2W4VEPZW5KRNPZU/bundle.json","state_url":"https://pith.science/pith/6WPLBLQG5UG2W4VEPZW5KRNPZU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6WPLBLQG5UG2W4VEPZW5KRNPZU/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":"2026-06-22T07:47:26Z","links":{"resolver":"https://pith.science/pith/6WPLBLQG5UG2W4VEPZW5KRNPZU","bundle":"https://pith.science/pith/6WPLBLQG5UG2W4VEPZW5KRNPZU/bundle.json","state":"https://pith.science/pith/6WPLBLQG5UG2W4VEPZW5KRNPZU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6WPLBLQG5UG2W4VEPZW5KRNPZU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:6WPLBLQG5UG2W4VEPZW5KRNPZU","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":"7036f31f944add93dae1cd6a9bbf0d9cea6279a02bc71acec3d894aaf1be68af","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-09-26T05:56:06Z","title_canon_sha256":"e808c5f56275af189a0ccf12c7b825e8b1a099087157bb30965b526f3ae25282"},"schema_version":"1.0","source":{"id":"1009.5065","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.5065","created_at":"2026-05-18T03:03:33Z"},{"alias_kind":"arxiv_version","alias_value":"1009.5065v1","created_at":"2026-05-18T03:03:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.5065","created_at":"2026-05-18T03:03:33Z"},{"alias_kind":"pith_short_12","alias_value":"6WPLBLQG5UG2","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"6WPLBLQG5UG2W4VE","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"6WPLBLQG","created_at":"2026-05-18T12:26:04Z"}],"graph_snapshots":[{"event_id":"sha256:7a051fad03f0560dd55a077f0f8731536236bc9ceadb13fc101710e5ba3088b2","target":"graph","created_at":"2026-05-18T03:03:33Z","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"},"paper":{"abstract_excerpt":"Let $\\Omega$ be an open subset of $\\Ri^d$ with $0\\in \\Omega$.\n  Further let $H_\\Omega=-\\sum^d_{i,j=1}\\partial_i\\,c_{ij}\\,\\partial_j$ be a second-order partial differential operator with domain $C_c^\\infty(\\Omega)$ where the coefficients $c_{ij}\\in W^{1,\\infty}_{\\rm loc}(\\bar\\Omega)$ are real, $c_{ij}=c_{ji}$ and the coefficient matrix $C=(c_{ij})$ satisfies bounds $0<C(x)\\leq c(|x|) I$ for all $x\\in \\Omega$. If \\[ \\int^\\infty_0ds\\,s^{d/2}\\,e^{-\\lambda\\,\\mu(s)^2}<\\infty \\] for some $\\lambda>0$ where $\\mu(s)=\\int^s_0dt\\,c(t)^{-1/2}$ then we establish that $H_\\Omega$ is $L_1$-unique, i.e.\\ it has","authors_text":"Adam Sikora, Derek W. Robinson","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-09-26T05:56:06Z","title":"$L_1$-uniqueness of degenerate elliptic operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.5065","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":"sha256:81ddc133a3a0ce3ab7ec4ad8a9e96f465200abc617e28f1e2c31794c8260ae7c","target":"record","created_at":"2026-05-18T03:03:33Z","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":"7036f31f944add93dae1cd6a9bbf0d9cea6279a02bc71acec3d894aaf1be68af","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-09-26T05:56:06Z","title_canon_sha256":"e808c5f56275af189a0ccf12c7b825e8b1a099087157bb30965b526f3ae25282"},"schema_version":"1.0","source":{"id":"1009.5065","kind":"arxiv","version":1}},"canonical_sha256":"f59eb0ae06ed0dab72a47e6dd545afcd27b6276325d2aa3c5e20cf85fcb8cba3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f59eb0ae06ed0dab72a47e6dd545afcd27b6276325d2aa3c5e20cf85fcb8cba3","first_computed_at":"2026-05-18T03:03:33.046430Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:03:33.046430Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jwdq4CJhrcpNk7CEn8yW1PvTlEnQliDPjbEp2hPiOltz6QHpBptNTNO98simoGrWkOWVy55KFnxD3J4B5U23BA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:03:33.046953Z","signed_message":"canonical_sha256_bytes"},"source_id":"1009.5065","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:81ddc133a3a0ce3ab7ec4ad8a9e96f465200abc617e28f1e2c31794c8260ae7c","sha256:7a051fad03f0560dd55a077f0f8731536236bc9ceadb13fc101710e5ba3088b2"],"state_sha256":"7427cf4078e96aee256451533cb2495dd17ecee21d44e47070112207daf4198e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eLaq4AqTzLWRRt8kXXj1BglrfHFzmVPC7d2MTZIXJm5qRTMbkhlvC0sKnoxkKRehDHRQN4u/2H/IDXkSLQ2eAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T07:47:26.794146Z","bundle_sha256":"bdcfe62289bea511e87245aeb52bdd4075f0aae0c45e95169842746829ca7ee0"}}