{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:P26QDMRXKDYK3SVDIVRHXZ4YXU","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":"6c396847f839da1cc2b0b405c7866edc2d3a26a20e4e1cb1de41850f6901de3b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-09-07T22:28:41Z","title_canon_sha256":"637e6ff490f86c2ab6bed956633d97f94e9d60c12fdcf713d62b7785d2b370be"},"schema_version":"1.0","source":{"id":"1709.02474","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.02474","created_at":"2026-05-18T00:35:45Z"},{"alias_kind":"arxiv_version","alias_value":"1709.02474v1","created_at":"2026-05-18T00:35:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.02474","created_at":"2026-05-18T00:35:45Z"},{"alias_kind":"pith_short_12","alias_value":"P26QDMRXKDYK","created_at":"2026-05-18T12:31:37Z"},{"alias_kind":"pith_short_16","alias_value":"P26QDMRXKDYK3SVD","created_at":"2026-05-18T12:31:37Z"},{"alias_kind":"pith_short_8","alias_value":"P26QDMRX","created_at":"2026-05-18T12:31:37Z"}],"graph_snapshots":[{"event_id":"sha256:3d9ce6548e563decbbabb1c010a86a683244256eeb374e24f40677f887c51171","target":"graph","created_at":"2026-05-18T00:35:45Z","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 prove the existence of a family of blow-up solutions of a mean field equation on sphere. The solutions blow up at four points where the minimum value of a potential energy function (involving the Green's function) is attained. The four blow-up points form a regular tetrahedron. Moreover, the solutions we build have a group of symmetry $T_d$ which is isomorphic to the symmetric group $S_4$. Other families of solutions can be similarly constructed with blow-up points at the vertices of equilateral triangles on a great circle or other inscribed platonic solids (cubes, octahedrons, icosahedrons","authors_text":"Changfeng Gui, Yeyao Hu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-09-07T22:28:41Z","title":"Non-axially symmetric solutions of a mean field equation on $\\mathbb{S}^2$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.02474","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":"sha256:2982162a4bc93837047302a3abfbd03fc07ad949f1ad10af0e5f615e821df966","target":"record","created_at":"2026-05-18T00:35:45Z","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":"6c396847f839da1cc2b0b405c7866edc2d3a26a20e4e1cb1de41850f6901de3b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-09-07T22:28:41Z","title_canon_sha256":"637e6ff490f86c2ab6bed956633d97f94e9d60c12fdcf713d62b7785d2b370be"},"schema_version":"1.0","source":{"id":"1709.02474","kind":"arxiv","version":1}},"canonical_sha256":"7ebd01b23750f0adcaa345627be798bd1a73cec30f9e34fa89e9c9bf51bccabc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7ebd01b23750f0adcaa345627be798bd1a73cec30f9e34fa89e9c9bf51bccabc","first_computed_at":"2026-05-18T00:35:45.687709Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:35:45.687709Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"L1IS1pFgNwYrStogNHQloK8IN4ut9wZNQZAZxEdkb9pG5+NL3u7gkb4rLYgXwQEjl2X2DncHiXW0HuFMXYNiDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:35:45.688310Z","signed_message":"canonical_sha256_bytes"},"source_id":"1709.02474","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2982162a4bc93837047302a3abfbd03fc07ad949f1ad10af0e5f615e821df966","sha256:3d9ce6548e563decbbabb1c010a86a683244256eeb374e24f40677f887c51171"],"state_sha256":"0f4597308cfe12702dc55117eddc41b471a918031a95cad8c6ec86cb2786461a"}