{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:ZYXZDE2D7UOLMKI7RNBSEWN3J3","short_pith_number":"pith:ZYXZDE2D","canonical_record":{"source":{"id":"1206.1514","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-06-07T14:46:24Z","cross_cats_sorted":[],"title_canon_sha256":"ba18b24ff6d3adf80456577f1e758d6e91d9ab0b9facde4e35d4b436ed2525ba","abstract_canon_sha256":"ad54c75b0be51c140f5bab6c7567ffc4446c2f20b25e0ab895aaf3b24c4e8bb9"},"schema_version":"1.0"},"canonical_sha256":"ce2f919343fd1cb6291f8b432259bb4ef1fe5cf88cc801d489248680218881a6","source":{"kind":"arxiv","id":"1206.1514","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.1514","created_at":"2026-05-18T03:53:19Z"},{"alias_kind":"arxiv_version","alias_value":"1206.1514v2","created_at":"2026-05-18T03:53:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.1514","created_at":"2026-05-18T03:53:19Z"},{"alias_kind":"pith_short_12","alias_value":"ZYXZDE2D7UOL","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_16","alias_value":"ZYXZDE2D7UOLMKI7","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_8","alias_value":"ZYXZDE2D","created_at":"2026-05-18T12:27:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:ZYXZDE2D7UOLMKI7RNBSEWN3J3","target":"record","payload":{"canonical_record":{"source":{"id":"1206.1514","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-06-07T14:46:24Z","cross_cats_sorted":[],"title_canon_sha256":"ba18b24ff6d3adf80456577f1e758d6e91d9ab0b9facde4e35d4b436ed2525ba","abstract_canon_sha256":"ad54c75b0be51c140f5bab6c7567ffc4446c2f20b25e0ab895aaf3b24c4e8bb9"},"schema_version":"1.0"},"canonical_sha256":"ce2f919343fd1cb6291f8b432259bb4ef1fe5cf88cc801d489248680218881a6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:53:19.514685Z","signature_b64":"c4K6G/eb6T3w0fO6umDIUWogmtmfpWkal6aZOMWem83bgwa115NmV3JZm0o+Qfn3t0j1fQWSMjnZVraLVFj4Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ce2f919343fd1cb6291f8b432259bb4ef1fe5cf88cc801d489248680218881a6","last_reissued_at":"2026-05-18T03:53:19.513861Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:53:19.513861Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1206.1514","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:53:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eiCQBxUp8gSYa9IaH8QJwaC4e7LJhbuNa5+sIQQRPipIw5lISMCW30dyKHXF1vJDtb3qIr5FCow0KPn975enBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T07:21:52.001647Z"},"content_sha256":"4135af1955c458aeba54495dae46c98d52e94efb411c5c7094b5eddd915c77f2","schema_version":"1.0","event_id":"sha256:4135af1955c458aeba54495dae46c98d52e94efb411c5c7094b5eddd915c77f2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:ZYXZDE2D7UOLMKI7RNBSEWN3J3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Champagne subregions with unavoidable bubbles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ivan Netuka, Wolfhard Hansen","submitted_at":"2012-06-07T14:46:24Z","abstract_excerpt":"A champagne subregion of a connected open set $U\\ne\\emptyset$ in $R^d$, $d\\ge 2$, is obtained omitting pairwise disjoint closed balls $\\bar B(x, r_x)$, $x\\in X$, the bubbles, where $X$ is a locally finite set in $U$. The union $A$ of these balls may be unavoidable, that is, Brownian motion, starting in $U\\setminus A$ and killed when leaving $U$, may hit $A$ almost surely or, equivalently, $A$ may have harmonic measure one for $U\\setminus A$.\n  Recent publications by Gardiner/Ghergu ($d\\ge 3$) and by Pres ($d=2$) give rather sharp answers to the question how small such a set $A$ may be, when $U"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.1514","kind":"arxiv","version":2},"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":"2026-05-18T03:53:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GoltQAZ3ctSjPNTWgXM9nek670GiQ4kkXIuQHzBSx99AIxPgLjQ27JmHV+ds70t3o7LDXjIsb87Tztcaw7y8CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T07:21:52.001972Z"},"content_sha256":"4d3bb6b3373b2fff96f55d143003af817963c28bc09db6169603794330a2209e","schema_version":"1.0","event_id":"sha256:4d3bb6b3373b2fff96f55d143003af817963c28bc09db6169603794330a2209e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZYXZDE2D7UOLMKI7RNBSEWN3J3/bundle.json","state_url":"https://pith.science/pith/ZYXZDE2D7UOLMKI7RNBSEWN3J3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZYXZDE2D7UOLMKI7RNBSEWN3J3/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":"2026-06-25T07:21:52Z","links":{"resolver":"https://pith.science/pith/ZYXZDE2D7UOLMKI7RNBSEWN3J3","bundle":"https://pith.science/pith/ZYXZDE2D7UOLMKI7RNBSEWN3J3/bundle.json","state":"https://pith.science/pith/ZYXZDE2D7UOLMKI7RNBSEWN3J3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZYXZDE2D7UOLMKI7RNBSEWN3J3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:ZYXZDE2D7UOLMKI7RNBSEWN3J3","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":"ad54c75b0be51c140f5bab6c7567ffc4446c2f20b25e0ab895aaf3b24c4e8bb9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-06-07T14:46:24Z","title_canon_sha256":"ba18b24ff6d3adf80456577f1e758d6e91d9ab0b9facde4e35d4b436ed2525ba"},"schema_version":"1.0","source":{"id":"1206.1514","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.1514","created_at":"2026-05-18T03:53:19Z"},{"alias_kind":"arxiv_version","alias_value":"1206.1514v2","created_at":"2026-05-18T03:53:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.1514","created_at":"2026-05-18T03:53:19Z"},{"alias_kind":"pith_short_12","alias_value":"ZYXZDE2D7UOL","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_16","alias_value":"ZYXZDE2D7UOLMKI7","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_8","alias_value":"ZYXZDE2D","created_at":"2026-05-18T12:27:30Z"}],"graph_snapshots":[{"event_id":"sha256:4d3bb6b3373b2fff96f55d143003af817963c28bc09db6169603794330a2209e","target":"graph","created_at":"2026-05-18T03:53:19Z","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":"A champagne subregion of a connected open set $U\\ne\\emptyset$ in $R^d$, $d\\ge 2$, is obtained omitting pairwise disjoint closed balls $\\bar B(x, r_x)$, $x\\in X$, the bubbles, where $X$ is a locally finite set in $U$. The union $A$ of these balls may be unavoidable, that is, Brownian motion, starting in $U\\setminus A$ and killed when leaving $U$, may hit $A$ almost surely or, equivalently, $A$ may have harmonic measure one for $U\\setminus A$.\n  Recent publications by Gardiner/Ghergu ($d\\ge 3$) and by Pres ($d=2$) give rather sharp answers to the question how small such a set $A$ may be, when $U","authors_text":"Ivan Netuka, Wolfhard Hansen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-06-07T14:46:24Z","title":"Champagne subregions with unavoidable bubbles"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.1514","kind":"arxiv","version":2},"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":"sha256:4135af1955c458aeba54495dae46c98d52e94efb411c5c7094b5eddd915c77f2","target":"record","created_at":"2026-05-18T03:53:19Z","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":"ad54c75b0be51c140f5bab6c7567ffc4446c2f20b25e0ab895aaf3b24c4e8bb9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-06-07T14:46:24Z","title_canon_sha256":"ba18b24ff6d3adf80456577f1e758d6e91d9ab0b9facde4e35d4b436ed2525ba"},"schema_version":"1.0","source":{"id":"1206.1514","kind":"arxiv","version":2}},"canonical_sha256":"ce2f919343fd1cb6291f8b432259bb4ef1fe5cf88cc801d489248680218881a6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ce2f919343fd1cb6291f8b432259bb4ef1fe5cf88cc801d489248680218881a6","first_computed_at":"2026-05-18T03:53:19.513861Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:53:19.513861Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"c4K6G/eb6T3w0fO6umDIUWogmtmfpWkal6aZOMWem83bgwa115NmV3JZm0o+Qfn3t0j1fQWSMjnZVraLVFj4Bw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:53:19.514685Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.1514","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4135af1955c458aeba54495dae46c98d52e94efb411c5c7094b5eddd915c77f2","sha256:4d3bb6b3373b2fff96f55d143003af817963c28bc09db6169603794330a2209e"],"state_sha256":"16b89e57c557b832d3139c234c634750bb8b762c31ee5ea9cdacbdb61000ced1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Qnbvdciec8tSkivPS7kQTFMXJudJ7+5gkLSmkyfgzeBEO1yImSZy2GOKYCsDX/1dXrYZ9hR/KesGO6c3Qd5MAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T07:21:52.003784Z","bundle_sha256":"07b7bc010f7f725d9797877e31d8fce8b6e12aaaa0d7e603379e79e5ea2145c8"}}