{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:2UPJKD4HNJG7JUS6LBTIM2RXTY","short_pith_number":"pith:2UPJKD4H","canonical_record":{"source":{"id":"2109.01697","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2021-09-03T18:31:19Z","cross_cats_sorted":["math-ph","math.CO","math.MP"],"title_canon_sha256":"ff310ea6c081d80ab5def812128a7189198f11752fb5fa8bc9409e7dd720fe1c","abstract_canon_sha256":"52276ed579df778883671a6edceb954faa5fdfaebb9838ce963c226eb0ca1480"},"schema_version":"1.0"},"canonical_sha256":"d51e950f876a4df4d25e5866866a379e00285b2f158115f05ede89c7a059c9d7","source":{"kind":"arxiv","id":"2109.01697","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2109.01697","created_at":"2026-07-05T06:16:58Z"},{"alias_kind":"arxiv_version","alias_value":"2109.01697v3","created_at":"2026-07-05T06:16:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2109.01697","created_at":"2026-07-05T06:16:58Z"},{"alias_kind":"pith_short_12","alias_value":"2UPJKD4HNJG7","created_at":"2026-07-05T06:16:58Z"},{"alias_kind":"pith_short_16","alias_value":"2UPJKD4HNJG7JUS6","created_at":"2026-07-05T06:16:58Z"},{"alias_kind":"pith_short_8","alias_value":"2UPJKD4H","created_at":"2026-07-05T06:16:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:2UPJKD4HNJG7JUS6LBTIM2RXTY","target":"record","payload":{"canonical_record":{"source":{"id":"2109.01697","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2021-09-03T18:31:19Z","cross_cats_sorted":["math-ph","math.CO","math.MP"],"title_canon_sha256":"ff310ea6c081d80ab5def812128a7189198f11752fb5fa8bc9409e7dd720fe1c","abstract_canon_sha256":"52276ed579df778883671a6edceb954faa5fdfaebb9838ce963c226eb0ca1480"},"schema_version":"1.0"},"canonical_sha256":"d51e950f876a4df4d25e5866866a379e00285b2f158115f05ede89c7a059c9d7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:16:58.644610Z","signature_b64":"ek27VOVHbPpwIEBvcI5CaJJNp30gAIn3DX+riKzrJ7d71YMxVmAuRDCWSbAivKkvX59gp2YvyGnP87v9VzlhDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d51e950f876a4df4d25e5866866a379e00285b2f158115f05ede89c7a059c9d7","last_reissued_at":"2026-07-05T06:16:58.644053Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:16:58.644053Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2109.01697","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T06:16:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SOITAAg21ZsWpQ/aK9w596mNlqKL+4kjZOKS3s0BErlj5c+bjeODqZysrJd6b8oT1NJykqXS7ZTm+XeboedCAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-13T16:14:25.027867Z"},"content_sha256":"762b3758f5d4f8cecb9b56fbd1d4098dfb5b2301d6624a1210cf81b8fa450f5d","schema_version":"1.0","event_id":"sha256:762b3758f5d4f8cecb9b56fbd1d4098dfb5b2301d6624a1210cf81b8fa450f5d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:2UPJKD4HNJG7JUS6LBTIM2RXTY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The double-bubble problem on the square lattice","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.CO","math.MP"],"primary_cat":"math.MG","authors_text":"Manuel Friedrich, Ulisse Stefanelli, Wojciech G\\'orny","submitted_at":"2021-09-03T18:31:19Z","abstract_excerpt":"We investigate minimal-perimeter configurations of two finite sets of points on the square lattice. This corresponds to a lattice version of the classical double-bubble problem. We give a detailed description of the fine geometry of minimisers and, in some parameter regime, we compute the optimal perimeter as a function of the size of the point sets. Moreover, we provide a sharp bound on the difference between two minimisers, which are generally not unique, and use it to rigorously identify their Wulff shape, as the size of the point sets scales up."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.01697","kind":"arxiv","version":3},"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/2109.01697/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":"2026-07-05T06:16:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vHIU6QlwwfxzHgbhePJwkhrpwg6D6+rXbl5E+n1jlPVMIUUnOo3f1TYRm+Rw4AIhPlnflWhUVx/RJSNqZ8IQAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-13T16:14:25.028238Z"},"content_sha256":"96b74690fbb42aefc51c870ac3207b910e56f34797aa4d72cd337eec90c81f25","schema_version":"1.0","event_id":"sha256:96b74690fbb42aefc51c870ac3207b910e56f34797aa4d72cd337eec90c81f25"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2UPJKD4HNJG7JUS6LBTIM2RXTY/bundle.json","state_url":"https://pith.science/pith/2UPJKD4HNJG7JUS6LBTIM2RXTY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2UPJKD4HNJG7JUS6LBTIM2RXTY/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":"2026-07-13T16:14:25Z","links":{"resolver":"https://pith.science/pith/2UPJKD4HNJG7JUS6LBTIM2RXTY","bundle":"https://pith.science/pith/2UPJKD4HNJG7JUS6LBTIM2RXTY/bundle.json","state":"https://pith.science/pith/2UPJKD4HNJG7JUS6LBTIM2RXTY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2UPJKD4HNJG7JUS6LBTIM2RXTY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:2UPJKD4HNJG7JUS6LBTIM2RXTY","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":"52276ed579df778883671a6edceb954faa5fdfaebb9838ce963c226eb0ca1480","cross_cats_sorted":["math-ph","math.CO","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2021-09-03T18:31:19Z","title_canon_sha256":"ff310ea6c081d80ab5def812128a7189198f11752fb5fa8bc9409e7dd720fe1c"},"schema_version":"1.0","source":{"id":"2109.01697","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2109.01697","created_at":"2026-07-05T06:16:58Z"},{"alias_kind":"arxiv_version","alias_value":"2109.01697v3","created_at":"2026-07-05T06:16:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2109.01697","created_at":"2026-07-05T06:16:58Z"},{"alias_kind":"pith_short_12","alias_value":"2UPJKD4HNJG7","created_at":"2026-07-05T06:16:58Z"},{"alias_kind":"pith_short_16","alias_value":"2UPJKD4HNJG7JUS6","created_at":"2026-07-05T06:16:58Z"},{"alias_kind":"pith_short_8","alias_value":"2UPJKD4H","created_at":"2026-07-05T06:16:58Z"}],"graph_snapshots":[{"event_id":"sha256:96b74690fbb42aefc51c870ac3207b910e56f34797aa4d72cd337eec90c81f25","target":"graph","created_at":"2026-07-05T06:16:58Z","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/2109.01697/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We investigate minimal-perimeter configurations of two finite sets of points on the square lattice. This corresponds to a lattice version of the classical double-bubble problem. We give a detailed description of the fine geometry of minimisers and, in some parameter regime, we compute the optimal perimeter as a function of the size of the point sets. Moreover, we provide a sharp bound on the difference between two minimisers, which are generally not unique, and use it to rigorously identify their Wulff shape, as the size of the point sets scales up.","authors_text":"Manuel Friedrich, Ulisse Stefanelli, Wojciech G\\'orny","cross_cats":["math-ph","math.CO","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2021-09-03T18:31:19Z","title":"The double-bubble problem on the square lattice"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.01697","kind":"arxiv","version":3},"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":"sha256:762b3758f5d4f8cecb9b56fbd1d4098dfb5b2301d6624a1210cf81b8fa450f5d","target":"record","created_at":"2026-07-05T06:16:58Z","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":"52276ed579df778883671a6edceb954faa5fdfaebb9838ce963c226eb0ca1480","cross_cats_sorted":["math-ph","math.CO","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2021-09-03T18:31:19Z","title_canon_sha256":"ff310ea6c081d80ab5def812128a7189198f11752fb5fa8bc9409e7dd720fe1c"},"schema_version":"1.0","source":{"id":"2109.01697","kind":"arxiv","version":3}},"canonical_sha256":"d51e950f876a4df4d25e5866866a379e00285b2f158115f05ede89c7a059c9d7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d51e950f876a4df4d25e5866866a379e00285b2f158115f05ede89c7a059c9d7","first_computed_at":"2026-07-05T06:16:58.644053Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:16:58.644053Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ek27VOVHbPpwIEBvcI5CaJJNp30gAIn3DX+riKzrJ7d71YMxVmAuRDCWSbAivKkvX59gp2YvyGnP87v9VzlhDg==","signature_status":"signed_v1","signed_at":"2026-07-05T06:16:58.644610Z","signed_message":"canonical_sha256_bytes"},"source_id":"2109.01697","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:762b3758f5d4f8cecb9b56fbd1d4098dfb5b2301d6624a1210cf81b8fa450f5d","sha256:96b74690fbb42aefc51c870ac3207b910e56f34797aa4d72cd337eec90c81f25"],"state_sha256":"c3f5e8c07f359d7151c108359d6d359d73239ce3b42c7b0b05893a70366448fe"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"F7/JlseYuNp7v9MdKGFBFx7URgeG1xxekXEIH0qxyWNz1IRPDALDZX4UKqneFM+++1laAWlxi7fipvNjLvVbBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-13T16:14:25.030344Z","bundle_sha256":"65de85372b8003a83fb8c9352bdb9457444a2cf322ac85c9cfb69611538eedeb"}}