{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:QXMCZBUXRJ6CPXYIXL4C6GQJDC","short_pith_number":"pith:QXMCZBUX","canonical_record":{"source":{"id":"2401.08063","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-01-16T02:46:40Z","cross_cats_sorted":["math.DG","math.MG"],"title_canon_sha256":"6751ceaf9ac033431664450b8213da87e7d7249d659279c93653fa9349bbaa00","abstract_canon_sha256":"b2366db66de12c2b4229975b04c11a747a3b57d45c34f55f78054fad4e19f788"},"schema_version":"1.0"},"canonical_sha256":"85d82c86978a7c27df08baf82f1a0918a04e5490b1d5236e00d78b6d541364fc","source":{"kind":"arxiv","id":"2401.08063","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2401.08063","created_at":"2026-07-05T11:12:16Z"},{"alias_kind":"arxiv_version","alias_value":"2401.08063v2","created_at":"2026-07-05T11:12:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2401.08063","created_at":"2026-07-05T11:12:16Z"},{"alias_kind":"pith_short_12","alias_value":"QXMCZBUXRJ6C","created_at":"2026-07-05T11:12:16Z"},{"alias_kind":"pith_short_16","alias_value":"QXMCZBUXRJ6CPXYI","created_at":"2026-07-05T11:12:16Z"},{"alias_kind":"pith_short_8","alias_value":"QXMCZBUX","created_at":"2026-07-05T11:12:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:QXMCZBUXRJ6CPXYIXL4C6GQJDC","target":"record","payload":{"canonical_record":{"source":{"id":"2401.08063","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-01-16T02:46:40Z","cross_cats_sorted":["math.DG","math.MG"],"title_canon_sha256":"6751ceaf9ac033431664450b8213da87e7d7249d659279c93653fa9349bbaa00","abstract_canon_sha256":"b2366db66de12c2b4229975b04c11a747a3b57d45c34f55f78054fad4e19f788"},"schema_version":"1.0"},"canonical_sha256":"85d82c86978a7c27df08baf82f1a0918a04e5490b1d5236e00d78b6d541364fc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:12:16.554469Z","signature_b64":"MN3FRKh4pegytxj1ajn9nF/WTLi/EUbwoW54ozABvU4McoqPQYbIpqbDLk4eiczPikMaFauL7QckTNqhOwx1Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"85d82c86978a7c27df08baf82f1a0918a04e5490b1d5236e00d78b6d541364fc","last_reissued_at":"2026-07-05T11:12:16.553977Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:12:16.553977Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2401.08063","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":"2026-07-05T11:12:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Yjr5uzJzfPikvE2nU7RGk+WMYv7YvUCKRPu3RJQxE3Aw9NiIEWZVoqKnzZI/1Lwt0N5QHYTiA8OnXerpRNEUCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T15:47:52.655556Z"},"content_sha256":"2d1b3c817693f8a07bb683d6d134b975718946dc953f1dceb6efe5ece6f7bf32","schema_version":"1.0","event_id":"sha256:2d1b3c817693f8a07bb683d6d134b975718946dc953f1dceb6efe5ece6f7bf32"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:QXMCZBUXRJ6CPXYIXL4C6GQJDC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An Infinite Double Bubble Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.MG"],"primary_cat":"math.AP","authors_text":"Lia Bronsard, Michael Novack","submitted_at":"2024-01-16T02:46:40Z","abstract_excerpt":"The classical double bubble theorem characterizes the minimizing partitions of $\\mathbb{R}^n$ into three chambers, two of which have prescribed finite volume. In this paper we prove a variant of the double bubble theorem in which two of the chambers have infinite volume. Such a configuration is an example of a (1,2)-cluster, or a partition of $\\mathbb{R}^n$ into three chambers, two of which have infinite volume and only one of which has finite volume. A $(1,2)$-cluster is locally minimizing with respect to a family of weights $\\{c_{jk}\\}$ if for any $B_r(0)$, it minimizes the interfacial energ"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2401.08063","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2401.08063/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T11:12:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"w4gjRoYFYbMKHX1DHMAPCdroUG2gKjotZcC9bAu8Ytigq5gDYH+tKC4H3thiCkwViTMsqy1vpgXRgUTjBAWuBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T15:47:52.657895Z"},"content_sha256":"f9d45e1b1d0ae826fccb9eb04ba0b8671d6abaf180bf2e1d9b2f3efb4d4067bd","schema_version":"1.0","event_id":"sha256:f9d45e1b1d0ae826fccb9eb04ba0b8671d6abaf180bf2e1d9b2f3efb4d4067bd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QXMCZBUXRJ6CPXYIXL4C6GQJDC/bundle.json","state_url":"https://pith.science/pith/QXMCZBUXRJ6CPXYIXL4C6GQJDC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QXMCZBUXRJ6CPXYIXL4C6GQJDC/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":"2026-08-11T15:47:52Z","links":{"resolver":"https://pith.science/pith/QXMCZBUXRJ6CPXYIXL4C6GQJDC","bundle":"https://pith.science/pith/QXMCZBUXRJ6CPXYIXL4C6GQJDC/bundle.json","state":"https://pith.science/pith/QXMCZBUXRJ6CPXYIXL4C6GQJDC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QXMCZBUXRJ6CPXYIXL4C6GQJDC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:QXMCZBUXRJ6CPXYIXL4C6GQJDC","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":"b2366db66de12c2b4229975b04c11a747a3b57d45c34f55f78054fad4e19f788","cross_cats_sorted":["math.DG","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-01-16T02:46:40Z","title_canon_sha256":"6751ceaf9ac033431664450b8213da87e7d7249d659279c93653fa9349bbaa00"},"schema_version":"1.0","source":{"id":"2401.08063","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2401.08063","created_at":"2026-07-05T11:12:16Z"},{"alias_kind":"arxiv_version","alias_value":"2401.08063v2","created_at":"2026-07-05T11:12:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2401.08063","created_at":"2026-07-05T11:12:16Z"},{"alias_kind":"pith_short_12","alias_value":"QXMCZBUXRJ6C","created_at":"2026-07-05T11:12:16Z"},{"alias_kind":"pith_short_16","alias_value":"QXMCZBUXRJ6CPXYI","created_at":"2026-07-05T11:12:16Z"},{"alias_kind":"pith_short_8","alias_value":"QXMCZBUX","created_at":"2026-07-05T11:12:16Z"}],"graph_snapshots":[{"event_id":"sha256:f9d45e1b1d0ae826fccb9eb04ba0b8671d6abaf180bf2e1d9b2f3efb4d4067bd","target":"graph","created_at":"2026-07-05T11:12:16Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2401.08063/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The classical double bubble theorem characterizes the minimizing partitions of $\\mathbb{R}^n$ into three chambers, two of which have prescribed finite volume. In this paper we prove a variant of the double bubble theorem in which two of the chambers have infinite volume. Such a configuration is an example of a (1,2)-cluster, or a partition of $\\mathbb{R}^n$ into three chambers, two of which have infinite volume and only one of which has finite volume. A $(1,2)$-cluster is locally minimizing with respect to a family of weights $\\{c_{jk}\\}$ if for any $B_r(0)$, it minimizes the interfacial energ","authors_text":"Lia Bronsard, Michael Novack","cross_cats":["math.DG","math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-01-16T02:46:40Z","title":"An Infinite Double Bubble Theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2401.08063","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":"sha256:2d1b3c817693f8a07bb683d6d134b975718946dc953f1dceb6efe5ece6f7bf32","target":"record","created_at":"2026-07-05T11:12:16Z","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":"b2366db66de12c2b4229975b04c11a747a3b57d45c34f55f78054fad4e19f788","cross_cats_sorted":["math.DG","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-01-16T02:46:40Z","title_canon_sha256":"6751ceaf9ac033431664450b8213da87e7d7249d659279c93653fa9349bbaa00"},"schema_version":"1.0","source":{"id":"2401.08063","kind":"arxiv","version":2}},"canonical_sha256":"85d82c86978a7c27df08baf82f1a0918a04e5490b1d5236e00d78b6d541364fc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"85d82c86978a7c27df08baf82f1a0918a04e5490b1d5236e00d78b6d541364fc","first_computed_at":"2026-07-05T11:12:16.553977Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:12:16.553977Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MN3FRKh4pegytxj1ajn9nF/WTLi/EUbwoW54ozABvU4McoqPQYbIpqbDLk4eiczPikMaFauL7QckTNqhOwx1Dg==","signature_status":"signed_v1","signed_at":"2026-07-05T11:12:16.554469Z","signed_message":"canonical_sha256_bytes"},"source_id":"2401.08063","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2d1b3c817693f8a07bb683d6d134b975718946dc953f1dceb6efe5ece6f7bf32","sha256:f9d45e1b1d0ae826fccb9eb04ba0b8671d6abaf180bf2e1d9b2f3efb4d4067bd"],"state_sha256":"5af53618f3c49cc595233c78845f2cf1bc77a05ba736bea9339be7d3f6a4f089"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Zh79zEfiq+ZDO2OND1wE3Y11swzJpITajYis5DU7CMfYLbkS+1GkNyvXTQBh4UBH+U9Ry3am9C0ATv6zYUYTBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-11T15:47:52.669086Z","bundle_sha256":"429afeb80e14169be7699249adf5ab791a78ace65c5ca82a1b94aae59750aff8"}}