{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:HVRF4OVBSXKQ7QNPVAQAMF5DIG","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":"04ee48dfdf1f7d40d05ad698de5e062c589848c470416c285d74595d6fcd4e2b","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2021-07-27T00:57:38Z","title_canon_sha256":"3b5b0758a248d7f1347da6900242f9b61f857dea9b75ce61c835cbb3c9145899"},"schema_version":"1.0","source":{"id":"2107.12535","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2107.12535","created_at":"2026-07-05T03:00:57Z"},{"alias_kind":"arxiv_version","alias_value":"2107.12535v1","created_at":"2026-07-05T03:00:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2107.12535","created_at":"2026-07-05T03:00:57Z"},{"alias_kind":"pith_short_12","alias_value":"HVRF4OVBSXKQ","created_at":"2026-07-05T03:00:57Z"},{"alias_kind":"pith_short_16","alias_value":"HVRF4OVBSXKQ7QNP","created_at":"2026-07-05T03:00:57Z"},{"alias_kind":"pith_short_8","alias_value":"HVRF4OVB","created_at":"2026-07-05T03:00:57Z"}],"graph_snapshots":[{"event_id":"sha256:c598fadb658fde651a1644300bd3720d1484246f3e9a18a7ec6bb41d4ba54b9e","target":"graph","created_at":"2026-07-05T03:00:57Z","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/2107.12535/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $ G=(V,E) $ be a connected finite graph and $ \\Delta $ the usual graph Laplacian. In this paper, we consider a generalized self-dual Chern-Simons equation on the graph $G$ \\begin{eqnarray}\\label{one1} \\Delta{u}=-\\lambda{e^{F(u)}[e^{F(u)}-1]^2}+4\\pi\\sum_{i=1}^{M}{\\delta_{p_{j}}}, \\end{eqnarray} where \\begin{equation} F(u)=\\left\\{\\begin{array}{l}\n  \\widetilde{F}(u), \\ \\quad u\\leq0,\n  0, \\quad \\quad \\quad u>0,\n  \\end{array} \\right. \\end{equation} $ \\widetilde{F}(u) $ satisfies $ u=1+{\\widetilde {F}(u)}-e^{\\widetilde {F}(u)} $, $ \\lambda>0 $, $M$ is any fixed positive integer, $ \\delta_{p_{j}}","authors_text":"Peirong Zhong, Yingshu L\\\"u","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2021-07-27T00:57:38Z","title":"Existence of solutions to a generalized self-dual Chern-Simons equation on graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2107.12535","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:aae8000919c921c87a103c1c5dc4b38f78c8cf44d6769d576a7d1e7624cb9f31","target":"record","created_at":"2026-07-05T03:00:57Z","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":"04ee48dfdf1f7d40d05ad698de5e062c589848c470416c285d74595d6fcd4e2b","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2021-07-27T00:57:38Z","title_canon_sha256":"3b5b0758a248d7f1347da6900242f9b61f857dea9b75ce61c835cbb3c9145899"},"schema_version":"1.0","source":{"id":"2107.12535","kind":"arxiv","version":1}},"canonical_sha256":"3d625e3aa195d50fc1afa8200617a341817d661ff7827e2a931806ddea3768f9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3d625e3aa195d50fc1afa8200617a341817d661ff7827e2a931806ddea3768f9","first_computed_at":"2026-07-05T03:00:57.235663Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:00:57.235663Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BvfEYh82Em9e7gt1JuD2sm8TINp9D1KcrpZfS0i1TZxklZQhbTquf19zwtP1r2SnD8fNx+o2PKYkuhRekSFUDg==","signature_status":"signed_v1","signed_at":"2026-07-05T03:00:57.236103Z","signed_message":"canonical_sha256_bytes"},"source_id":"2107.12535","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:aae8000919c921c87a103c1c5dc4b38f78c8cf44d6769d576a7d1e7624cb9f31","sha256:c598fadb658fde651a1644300bd3720d1484246f3e9a18a7ec6bb41d4ba54b9e"],"state_sha256":"d056dc880c61f9e6ebec8bc105d8e57fb0dbaf6bc03db3a65f2c1bde68241fd1"}