{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:HZS6AYCBZPCOTKF3BGCATGWY7E","short_pith_number":"pith:HZS6AYCB","canonical_record":{"source":{"id":"1210.5098","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-10-18T11:34:55Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"1954ec61ea28529c30af7eb6dee50324d57c06e48c147b4306f9901a9979dcac","abstract_canon_sha256":"85c65c50dd54d449422d923154e97bfaa3caeaefc59db2b6a4608355e1bd5941"},"schema_version":"1.0"},"canonical_sha256":"3e65e06041cbc4e9a8bb0984099ad8f917dd625f1d9040d1a065232986f40dde","source":{"kind":"arxiv","id":"1210.5098","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.5098","created_at":"2026-05-18T03:42:54Z"},{"alias_kind":"arxiv_version","alias_value":"1210.5098v1","created_at":"2026-05-18T03:42:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.5098","created_at":"2026-05-18T03:42:54Z"},{"alias_kind":"pith_short_12","alias_value":"HZS6AYCBZPCO","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_16","alias_value":"HZS6AYCBZPCOTKF3","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_8","alias_value":"HZS6AYCB","created_at":"2026-05-18T12:27:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:HZS6AYCBZPCOTKF3BGCATGWY7E","target":"record","payload":{"canonical_record":{"source":{"id":"1210.5098","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-10-18T11:34:55Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"1954ec61ea28529c30af7eb6dee50324d57c06e48c147b4306f9901a9979dcac","abstract_canon_sha256":"85c65c50dd54d449422d923154e97bfaa3caeaefc59db2b6a4608355e1bd5941"},"schema_version":"1.0"},"canonical_sha256":"3e65e06041cbc4e9a8bb0984099ad8f917dd625f1d9040d1a065232986f40dde","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:42:54.280627Z","signature_b64":"Mtf3AptvUnUhQEXJFmEP7GF6wVpADPLg9ZbTANog/VSZDVgB3a+t4XFeDPVthxj6oRH5DOZaN9uADhYgwpdTAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3e65e06041cbc4e9a8bb0984099ad8f917dd625f1d9040d1a065232986f40dde","last_reissued_at":"2026-05-18T03:42:54.279824Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:42:54.279824Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1210.5098","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":"2026-05-18T03:42:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GmIR1SCplAEHQSO9tuWmMBA0qpVdVfm95rqvuCGl7/ni0J6uLwYP3M+94yu8Po1lUvwev15pxZIVittxZ8C4DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T16:33:43.143288Z"},"content_sha256":"c011c480696fc55a1606de9bcebcd3bac95efff8144e7cde625109418600502a","schema_version":"1.0","event_id":"sha256:c011c480696fc55a1606de9bcebcd3bac95efff8144e7cde625109418600502a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:HZS6AYCBZPCOTKF3BGCATGWY7E","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Gamma-limit of the two-dimensional Ohta-Kawasaki energy. II. Droplet arrangement at the sharp interface level via the renormalized energy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Cyrill B. Muratov, Dorian Goldman, Sylvia Serfaty","submitted_at":"2012-10-18T11:34:55Z","abstract_excerpt":"This is the second in a series of papers in which we derive a $\\Gamma$-expansion for the two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion known as the Ohta-Kawasaki model in connection with diblock copolymer systems. In this model, two phases appear, which interact via a nonlocal Coulomb type energy. Here we focus on the sharp interface version of this energy in the regime where one of the phases has very small volume fraction, thus creating small \"droplets\" of the minority phase in a \"sea\" of the majority phase. In our previous paper, we computed the $\\Gamma$-limit of t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.5098","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":"2026-05-18T03:42:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AAyBs50Cvu90p+6xdg3EYqyTZNeXoQNRFtY1p2OKI97rinXF+DW7Xb86i4w77umYfQoDV0aEfCuPCNjmM9jUBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T16:33:43.143897Z"},"content_sha256":"06fdf5da9bb064d10172b155d11b7f65c2c092c8fafb86468ee872583e51df3f","schema_version":"1.0","event_id":"sha256:06fdf5da9bb064d10172b155d11b7f65c2c092c8fafb86468ee872583e51df3f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HZS6AYCBZPCOTKF3BGCATGWY7E/bundle.json","state_url":"https://pith.science/pith/HZS6AYCBZPCOTKF3BGCATGWY7E/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HZS6AYCBZPCOTKF3BGCATGWY7E/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":"2026-05-26T16:33:43Z","links":{"resolver":"https://pith.science/pith/HZS6AYCBZPCOTKF3BGCATGWY7E","bundle":"https://pith.science/pith/HZS6AYCBZPCOTKF3BGCATGWY7E/bundle.json","state":"https://pith.science/pith/HZS6AYCBZPCOTKF3BGCATGWY7E/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HZS6AYCBZPCOTKF3BGCATGWY7E/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:HZS6AYCBZPCOTKF3BGCATGWY7E","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":"85c65c50dd54d449422d923154e97bfaa3caeaefc59db2b6a4608355e1bd5941","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-10-18T11:34:55Z","title_canon_sha256":"1954ec61ea28529c30af7eb6dee50324d57c06e48c147b4306f9901a9979dcac"},"schema_version":"1.0","source":{"id":"1210.5098","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.5098","created_at":"2026-05-18T03:42:54Z"},{"alias_kind":"arxiv_version","alias_value":"1210.5098v1","created_at":"2026-05-18T03:42:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.5098","created_at":"2026-05-18T03:42:54Z"},{"alias_kind":"pith_short_12","alias_value":"HZS6AYCBZPCO","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_16","alias_value":"HZS6AYCBZPCOTKF3","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_8","alias_value":"HZS6AYCB","created_at":"2026-05-18T12:27:09Z"}],"graph_snapshots":[{"event_id":"sha256:06fdf5da9bb064d10172b155d11b7f65c2c092c8fafb86468ee872583e51df3f","target":"graph","created_at":"2026-05-18T03:42:54Z","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":"This is the second in a series of papers in which we derive a $\\Gamma$-expansion for the two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion known as the Ohta-Kawasaki model in connection with diblock copolymer systems. In this model, two phases appear, which interact via a nonlocal Coulomb type energy. Here we focus on the sharp interface version of this energy in the regime where one of the phases has very small volume fraction, thus creating small \"droplets\" of the minority phase in a \"sea\" of the majority phase. In our previous paper, we computed the $\\Gamma$-limit of t","authors_text":"Cyrill B. Muratov, Dorian Goldman, Sylvia Serfaty","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-10-18T11:34:55Z","title":"The Gamma-limit of the two-dimensional Ohta-Kawasaki energy. II. Droplet arrangement at the sharp interface level via the renormalized energy"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.5098","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:c011c480696fc55a1606de9bcebcd3bac95efff8144e7cde625109418600502a","target":"record","created_at":"2026-05-18T03:42:54Z","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":"85c65c50dd54d449422d923154e97bfaa3caeaefc59db2b6a4608355e1bd5941","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-10-18T11:34:55Z","title_canon_sha256":"1954ec61ea28529c30af7eb6dee50324d57c06e48c147b4306f9901a9979dcac"},"schema_version":"1.0","source":{"id":"1210.5098","kind":"arxiv","version":1}},"canonical_sha256":"3e65e06041cbc4e9a8bb0984099ad8f917dd625f1d9040d1a065232986f40dde","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3e65e06041cbc4e9a8bb0984099ad8f917dd625f1d9040d1a065232986f40dde","first_computed_at":"2026-05-18T03:42:54.279824Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:42:54.279824Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Mtf3AptvUnUhQEXJFmEP7GF6wVpADPLg9ZbTANog/VSZDVgB3a+t4XFeDPVthxj6oRH5DOZaN9uADhYgwpdTAw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:42:54.280627Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.5098","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c011c480696fc55a1606de9bcebcd3bac95efff8144e7cde625109418600502a","sha256:06fdf5da9bb064d10172b155d11b7f65c2c092c8fafb86468ee872583e51df3f"],"state_sha256":"c884ca24d1908c7b97f51c06ab14f31c0fe0b3a844316fe9759dc891a3f7770e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Vgr92/OBgw95+sCg0QB2MCtx6tgfehR3WD7YX8EM51M2LtBsBr2Q0PL3LK25X8D1pkXc8pNlpZLAcyaAufNRBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T16:33:43.147087Z","bundle_sha256":"ba4a031062eb9e82b5bb6a9acef6135cf3823b8c5512177864a750e8a153c1a6"}}