{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:PRTLFMDFH4NHDU6WSDIZYSFGBJ","short_pith_number":"pith:PRTLFMDF","canonical_record":{"source":{"id":"1007.1725","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-07-10T14:55:51Z","cross_cats_sorted":[],"title_canon_sha256":"ecbe857055abd78e4128c8527dd32a3be34e56579c24fe669562ee6fadba76c8","abstract_canon_sha256":"2c8420a71deeb60fc255894adb0d5ace6feb1436e64184f31b25519ec1b28d5b"},"schema_version":"1.0"},"canonical_sha256":"7c66b2b0653f1a71d3d690d19c48a60a40fa5a439351bf24d08aebec980892b3","source":{"kind":"arxiv","id":"1007.1725","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.1725","created_at":"2026-05-18T04:24:54Z"},{"alias_kind":"arxiv_version","alias_value":"1007.1725v3","created_at":"2026-05-18T04:24:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.1725","created_at":"2026-05-18T04:24:54Z"},{"alias_kind":"pith_short_12","alias_value":"PRTLFMDFH4NH","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_16","alias_value":"PRTLFMDFH4NHDU6W","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_8","alias_value":"PRTLFMDF","created_at":"2026-05-18T12:26:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:PRTLFMDFH4NHDU6WSDIZYSFGBJ","target":"record","payload":{"canonical_record":{"source":{"id":"1007.1725","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-07-10T14:55:51Z","cross_cats_sorted":[],"title_canon_sha256":"ecbe857055abd78e4128c8527dd32a3be34e56579c24fe669562ee6fadba76c8","abstract_canon_sha256":"2c8420a71deeb60fc255894adb0d5ace6feb1436e64184f31b25519ec1b28d5b"},"schema_version":"1.0"},"canonical_sha256":"7c66b2b0653f1a71d3d690d19c48a60a40fa5a439351bf24d08aebec980892b3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:24:54.487573Z","signature_b64":"GRULHQq6eSpI4l5lQjSN7rLfzbcg9vKLQHDOyCjwZn+5sZ5c4kEdWcGOKl0eILCNKW/UcpNJ8aYT7DBkJl3mBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7c66b2b0653f1a71d3d690d19c48a60a40fa5a439351bf24d08aebec980892b3","last_reissued_at":"2026-05-18T04:24:54.487060Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:24:54.487060Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1007.1725","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":"2026-05-18T04:24:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Qw6OVtfrMFsFIGNx0psR5PcWfDHbLqvtPHJzojXAPCryIoJarm7SQwp5xhC5MvW+rjsFiAY/wHE5qTxi7xuPCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T22:10:09.816307Z"},"content_sha256":"e1622ee34595142dc1a23ed3d276dd4da2208a8a55cf8675f58287311a2daf43","schema_version":"1.0","event_id":"sha256:e1622ee34595142dc1a23ed3d276dd4da2208a8a55cf8675f58287311a2daf43"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:PRTLFMDFH4NHDU6WSDIZYSFGBJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"$\\Gamma$-convergence for nonlocal phase transitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Enrico Valdinoci, Ovidiu Savin","submitted_at":"2010-07-10T14:55:51Z","abstract_excerpt":"We discuss the $\\Gamma$-convergence, under the appropriate scaling, of the energy functional $$ \\|u\\|_{H^s(\\Omega)}^2+\\int_\\Omega W(u)dx,$$ with $s \\in (0,1)$, where $\\|u\\|_{H^s(\\Omega)}$ denotes the total contribution from $\\Omega$ in the $H^s$ norm of $u$, and $W$ is a double-well potential.\n  When $s\\in [1/2,\\,1)$, we show that the energy $\\Gamma$-converges to the classical minimal surface functional -- while, when $s\\in(0,\\,1/2)$, it is easy to see that the functional $\\Gamma$-converges to the nonlocal minimal surface functional."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.1725","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":""},"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":"2026-05-18T04:24:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5U6avPVQjD7RogYJSwagOA8ANTp6qfNzXYyUE45u+SkMn2Yjo23ogaXk1yP6/+SaHAZZIljmyyjIP48kXvF7AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T22:10:09.816660Z"},"content_sha256":"e3d0692fe457c09cbf89ee830a9057a293b239f399c19131f27fe48e04ea763d","schema_version":"1.0","event_id":"sha256:e3d0692fe457c09cbf89ee830a9057a293b239f399c19131f27fe48e04ea763d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PRTLFMDFH4NHDU6WSDIZYSFGBJ/bundle.json","state_url":"https://pith.science/pith/PRTLFMDFH4NHDU6WSDIZYSFGBJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PRTLFMDFH4NHDU6WSDIZYSFGBJ/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":"2026-06-08T22:10:09Z","links":{"resolver":"https://pith.science/pith/PRTLFMDFH4NHDU6WSDIZYSFGBJ","bundle":"https://pith.science/pith/PRTLFMDFH4NHDU6WSDIZYSFGBJ/bundle.json","state":"https://pith.science/pith/PRTLFMDFH4NHDU6WSDIZYSFGBJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PRTLFMDFH4NHDU6WSDIZYSFGBJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:PRTLFMDFH4NHDU6WSDIZYSFGBJ","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":"2c8420a71deeb60fc255894adb0d5ace6feb1436e64184f31b25519ec1b28d5b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-07-10T14:55:51Z","title_canon_sha256":"ecbe857055abd78e4128c8527dd32a3be34e56579c24fe669562ee6fadba76c8"},"schema_version":"1.0","source":{"id":"1007.1725","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.1725","created_at":"2026-05-18T04:24:54Z"},{"alias_kind":"arxiv_version","alias_value":"1007.1725v3","created_at":"2026-05-18T04:24:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.1725","created_at":"2026-05-18T04:24:54Z"},{"alias_kind":"pith_short_12","alias_value":"PRTLFMDFH4NH","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_16","alias_value":"PRTLFMDFH4NHDU6W","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_8","alias_value":"PRTLFMDF","created_at":"2026-05-18T12:26:12Z"}],"graph_snapshots":[{"event_id":"sha256:e3d0692fe457c09cbf89ee830a9057a293b239f399c19131f27fe48e04ea763d","target":"graph","created_at":"2026-05-18T04:24:54Z","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":"We discuss the $\\Gamma$-convergence, under the appropriate scaling, of the energy functional $$ \\|u\\|_{H^s(\\Omega)}^2+\\int_\\Omega W(u)dx,$$ with $s \\in (0,1)$, where $\\|u\\|_{H^s(\\Omega)}$ denotes the total contribution from $\\Omega$ in the $H^s$ norm of $u$, and $W$ is a double-well potential.\n  When $s\\in [1/2,\\,1)$, we show that the energy $\\Gamma$-converges to the classical minimal surface functional -- while, when $s\\in(0,\\,1/2)$, it is easy to see that the functional $\\Gamma$-converges to the nonlocal minimal surface functional.","authors_text":"Enrico Valdinoci, Ovidiu Savin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-07-10T14:55:51Z","title":"$\\Gamma$-convergence for nonlocal phase transitions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.1725","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:e1622ee34595142dc1a23ed3d276dd4da2208a8a55cf8675f58287311a2daf43","target":"record","created_at":"2026-05-18T04:24:54Z","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":"2c8420a71deeb60fc255894adb0d5ace6feb1436e64184f31b25519ec1b28d5b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-07-10T14:55:51Z","title_canon_sha256":"ecbe857055abd78e4128c8527dd32a3be34e56579c24fe669562ee6fadba76c8"},"schema_version":"1.0","source":{"id":"1007.1725","kind":"arxiv","version":3}},"canonical_sha256":"7c66b2b0653f1a71d3d690d19c48a60a40fa5a439351bf24d08aebec980892b3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7c66b2b0653f1a71d3d690d19c48a60a40fa5a439351bf24d08aebec980892b3","first_computed_at":"2026-05-18T04:24:54.487060Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:24:54.487060Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GRULHQq6eSpI4l5lQjSN7rLfzbcg9vKLQHDOyCjwZn+5sZ5c4kEdWcGOKl0eILCNKW/UcpNJ8aYT7DBkJl3mBw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:24:54.487573Z","signed_message":"canonical_sha256_bytes"},"source_id":"1007.1725","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e1622ee34595142dc1a23ed3d276dd4da2208a8a55cf8675f58287311a2daf43","sha256:e3d0692fe457c09cbf89ee830a9057a293b239f399c19131f27fe48e04ea763d"],"state_sha256":"b4c5a2ad16ccf9e19d4343edb94ffb8785e5cd5364fd10666589f7d826bb58ea"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7EV2hVX0UwDdwruVjzgZ88OPHiDZhwnbP75tcSyGJSUhz/k5jTN38FOgO7Df79rJrkGRQJpVkNO4dL3SpitVAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T22:10:09.818484Z","bundle_sha256":"06fc18f28030faa0693f60c9f3ccfd04b0a0085b771a537342ad0ab0b6356dc0"}}