{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:CHYJO2ZPCLBIU4DBMRVRXCNE63","short_pith_number":"pith:CHYJO2ZP","canonical_record":{"source":{"id":"1812.08504","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-12-20T12:00:25Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"00a68d1e2cc5a2b2f705457cca0899c9ea0ad69bd2506a1cc714f49f99d42026","abstract_canon_sha256":"bbe3f88abc69243b4b9f23e24f517be9cc534c533969f04ac4dfcfbbb6c44791"},"schema_version":"1.0"},"canonical_sha256":"11f0976b2f12c28a7061646b1b89a4f6f12066da5e53cb9ff15573a8c6c5e507","source":{"kind":"arxiv","id":"1812.08504","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.08504","created_at":"2026-07-05T00:57:43Z"},{"alias_kind":"arxiv_version","alias_value":"1812.08504v2","created_at":"2026-07-05T00:57:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.08504","created_at":"2026-07-05T00:57:43Z"},{"alias_kind":"pith_short_12","alias_value":"CHYJO2ZPCLBI","created_at":"2026-07-05T00:57:43Z"},{"alias_kind":"pith_short_16","alias_value":"CHYJO2ZPCLBIU4DB","created_at":"2026-07-05T00:57:43Z"},{"alias_kind":"pith_short_8","alias_value":"CHYJO2ZP","created_at":"2026-07-05T00:57:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:CHYJO2ZPCLBIU4DBMRVRXCNE63","target":"record","payload":{"canonical_record":{"source":{"id":"1812.08504","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-12-20T12:00:25Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"00a68d1e2cc5a2b2f705457cca0899c9ea0ad69bd2506a1cc714f49f99d42026","abstract_canon_sha256":"bbe3f88abc69243b4b9f23e24f517be9cc534c533969f04ac4dfcfbbb6c44791"},"schema_version":"1.0"},"canonical_sha256":"11f0976b2f12c28a7061646b1b89a4f6f12066da5e53cb9ff15573a8c6c5e507","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:57:43.637808Z","signature_b64":"aEvD8+Nsfz5ya6UeHBIFpXIarQEE3BeRvA6X4A54+9lckjsLnN1LE2IqeQ+i0ZDRwb1wd3grXIkiP0xqTpbhCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"11f0976b2f12c28a7061646b1b89a4f6f12066da5e53cb9ff15573a8c6c5e507","last_reissued_at":"2026-07-05T00:57:43.637231Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:57:43.637231Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1812.08504","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":"2026-07-05T00:57:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KTOoNDtG0IxvxK6hjv7jUiqpPQ97qPc9KDpHiytF31qA0e4jxvE5A62lWED/IvFnKyaF5Ep/htxKjLUOtgnPAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T09:20:56.446261Z"},"content_sha256":"93c00297a5ae6f4be78db6dd9b4891f97ba527692ddd0941fc5580ac626abd1d","schema_version":"1.0","event_id":"sha256:93c00297a5ae6f4be78db6dd9b4891f97ba527692ddd0941fc5580ac626abd1d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:CHYJO2ZPCLBIU4DBMRVRXCNE63","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Generic uniqueness of expanders with vanishing relative entropy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Alix Deruelle, Felix Schulze","submitted_at":"2018-12-20T12:00:25Z","abstract_excerpt":"We define a relative entropy for two expanding solutions to mean curvature flow of hypersurfaces, asymptotic to the same cone at infinity. Adapting work of White and using recent results of Bernstein and Bernstein-Wang, we show that expanders with vanishing relative entropy are unique in a generic sense. This also implies that generically locally entropy minimising expanders are unique."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.08504","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/1812.08504/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":"2026-07-05T00:57:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"88bneTecY7d7WqpjSy64pLEOI6OVaMIDhYLgz3E9M7lnU/dut339MYMK/Y3EJiHi+6fjQdaswWRAq/dATQsxDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T09:20:56.446589Z"},"content_sha256":"f3037d292cb4cbcf9baf6793edc5206c70db49c812b4a5c7a1effe9bdeca1337","schema_version":"1.0","event_id":"sha256:f3037d292cb4cbcf9baf6793edc5206c70db49c812b4a5c7a1effe9bdeca1337"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CHYJO2ZPCLBIU4DBMRVRXCNE63/bundle.json","state_url":"https://pith.science/pith/CHYJO2ZPCLBIU4DBMRVRXCNE63/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CHYJO2ZPCLBIU4DBMRVRXCNE63/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":"2026-08-14T09:20:56Z","links":{"resolver":"https://pith.science/pith/CHYJO2ZPCLBIU4DBMRVRXCNE63","bundle":"https://pith.science/pith/CHYJO2ZPCLBIU4DBMRVRXCNE63/bundle.json","state":"https://pith.science/pith/CHYJO2ZPCLBIU4DBMRVRXCNE63/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CHYJO2ZPCLBIU4DBMRVRXCNE63/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:CHYJO2ZPCLBIU4DBMRVRXCNE63","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":"bbe3f88abc69243b4b9f23e24f517be9cc534c533969f04ac4dfcfbbb6c44791","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-12-20T12:00:25Z","title_canon_sha256":"00a68d1e2cc5a2b2f705457cca0899c9ea0ad69bd2506a1cc714f49f99d42026"},"schema_version":"1.0","source":{"id":"1812.08504","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.08504","created_at":"2026-07-05T00:57:43Z"},{"alias_kind":"arxiv_version","alias_value":"1812.08504v2","created_at":"2026-07-05T00:57:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.08504","created_at":"2026-07-05T00:57:43Z"},{"alias_kind":"pith_short_12","alias_value":"CHYJO2ZPCLBI","created_at":"2026-07-05T00:57:43Z"},{"alias_kind":"pith_short_16","alias_value":"CHYJO2ZPCLBIU4DB","created_at":"2026-07-05T00:57:43Z"},{"alias_kind":"pith_short_8","alias_value":"CHYJO2ZP","created_at":"2026-07-05T00:57:43Z"}],"graph_snapshots":[{"event_id":"sha256:f3037d292cb4cbcf9baf6793edc5206c70db49c812b4a5c7a1effe9bdeca1337","target":"graph","created_at":"2026-07-05T00:57:43Z","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/1812.08504/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We define a relative entropy for two expanding solutions to mean curvature flow of hypersurfaces, asymptotic to the same cone at infinity. Adapting work of White and using recent results of Bernstein and Bernstein-Wang, we show that expanders with vanishing relative entropy are unique in a generic sense. This also implies that generically locally entropy minimising expanders are unique.","authors_text":"Alix Deruelle, Felix Schulze","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-12-20T12:00:25Z","title":"Generic uniqueness of expanders with vanishing relative entropy"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.08504","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":"sha256:93c00297a5ae6f4be78db6dd9b4891f97ba527692ddd0941fc5580ac626abd1d","target":"record","created_at":"2026-07-05T00:57:43Z","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":"bbe3f88abc69243b4b9f23e24f517be9cc534c533969f04ac4dfcfbbb6c44791","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-12-20T12:00:25Z","title_canon_sha256":"00a68d1e2cc5a2b2f705457cca0899c9ea0ad69bd2506a1cc714f49f99d42026"},"schema_version":"1.0","source":{"id":"1812.08504","kind":"arxiv","version":2}},"canonical_sha256":"11f0976b2f12c28a7061646b1b89a4f6f12066da5e53cb9ff15573a8c6c5e507","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"11f0976b2f12c28a7061646b1b89a4f6f12066da5e53cb9ff15573a8c6c5e507","first_computed_at":"2026-07-05T00:57:43.637231Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:57:43.637231Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aEvD8+Nsfz5ya6UeHBIFpXIarQEE3BeRvA6X4A54+9lckjsLnN1LE2IqeQ+i0ZDRwb1wd3grXIkiP0xqTpbhCw==","signature_status":"signed_v1","signed_at":"2026-07-05T00:57:43.637808Z","signed_message":"canonical_sha256_bytes"},"source_id":"1812.08504","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:93c00297a5ae6f4be78db6dd9b4891f97ba527692ddd0941fc5580ac626abd1d","sha256:f3037d292cb4cbcf9baf6793edc5206c70db49c812b4a5c7a1effe9bdeca1337"],"state_sha256":"ed475fbfc1951119cbf9ff3fe4d4bdf28fe1474f2e4a426dd116488b5383867d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HX6+aNJZdwzKRffoSi4Cdpo+9wc+UBYy22+SMxWwWrXKgSGDCPN171r22fHI+5rV7YTytc+dhLyAygXjCmBjAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-14T09:20:56.449742Z","bundle_sha256":"3f44d258d8644dcc49fdfbf19318e293d8a3986f9924326d174447692aaea244"}}