{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:QTUGZ6VT5VBTSLS36WFPE3JMJL","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":"459d541af2bed0a61b91732447a0a4340c67ea3788346065d0ab60e768485cfd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-05-05T16:15:57Z","title_canon_sha256":"eba6b31f8068d4b27625d1dbcbf00316865a5ec880c5cecef3787364370d7b8e"},"schema_version":"1.0","source":{"id":"1405.0944","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1405.0944","created_at":"2026-05-18T02:45:14Z"},{"alias_kind":"arxiv_version","alias_value":"1405.0944v2","created_at":"2026-05-18T02:45:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.0944","created_at":"2026-05-18T02:45:14Z"},{"alias_kind":"pith_short_12","alias_value":"QTUGZ6VT5VBT","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_16","alias_value":"QTUGZ6VT5VBTSLS3","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_8","alias_value":"QTUGZ6VT","created_at":"2026-05-18T12:28:46Z"}],"graph_snapshots":[{"event_id":"sha256:8df1d50917db6009aa984c0a1cb7c947bdd4da075441a9135fe022945cd8c825","target":"graph","created_at":"2026-05-18T02:45:14Z","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 review and give elementary proofs of Liouville type properties of harmonic and subharmonic functions in the plane endowed with a complete Riemannian metric, and  prove a gap theorem for the possible growth of harmonic functions when this metric has nonnegative Gaussian curvature.","authors_text":"Jean C. Cortissoz","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-05-05T16:15:57Z","title":"A Note on Harmonic Functions on surfaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.0944","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:e63af7e43fd5b590ad929eb5eb7d0f26ee4625e1073902950e26bd8507caeff2","target":"record","created_at":"2026-05-18T02:45:14Z","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":"459d541af2bed0a61b91732447a0a4340c67ea3788346065d0ab60e768485cfd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-05-05T16:15:57Z","title_canon_sha256":"eba6b31f8068d4b27625d1dbcbf00316865a5ec880c5cecef3787364370d7b8e"},"schema_version":"1.0","source":{"id":"1405.0944","kind":"arxiv","version":2}},"canonical_sha256":"84e86cfab3ed43392e5bf58af26d2c4afa6f31308d2b58253abf3ea828a7f205","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"84e86cfab3ed43392e5bf58af26d2c4afa6f31308d2b58253abf3ea828a7f205","first_computed_at":"2026-05-18T02:45:14.093496Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:45:14.093496Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CH0bhRbZWf3knG8ngD6U7xwsV+U94g19rMXcNCzzfrTWvTtJ+2r+dtS4+OR6N632h6y8Fqxt9MG7VnlpwnDnCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:45:14.094120Z","signed_message":"canonical_sha256_bytes"},"source_id":"1405.0944","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e63af7e43fd5b590ad929eb5eb7d0f26ee4625e1073902950e26bd8507caeff2","sha256:8df1d50917db6009aa984c0a1cb7c947bdd4da075441a9135fe022945cd8c825"],"state_sha256":"1e5cc29cca5bdce4118b3f3f74c3e3d25fab392aa1d1e66466511ac121208a9e"}