{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:NBJQ3ZA4EXQM4LPP24PRQLD4YF","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":"50072a95c1821e227ff5ef47fdf8604ea6d23ea5392df53506307d4871ccfe96","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-06-09T19:52:43Z","title_canon_sha256":"457ea712c5bec771cc82c8f15ed647d3818d7cc0d46af06d2af9107d472837be"},"schema_version":"1.0","source":{"id":"2506.08187","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.08187","created_at":"2026-07-05T11:18:46Z"},{"alias_kind":"arxiv_version","alias_value":"2506.08187v1","created_at":"2026-07-05T11:18:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.08187","created_at":"2026-07-05T11:18:46Z"},{"alias_kind":"pith_short_12","alias_value":"NBJQ3ZA4EXQM","created_at":"2026-07-05T11:18:46Z"},{"alias_kind":"pith_short_16","alias_value":"NBJQ3ZA4EXQM4LPP","created_at":"2026-07-05T11:18:46Z"},{"alias_kind":"pith_short_8","alias_value":"NBJQ3ZA4","created_at":"2026-07-05T11:18:46Z"}],"graph_snapshots":[{"event_id":"sha256:431aa15eed5f808bc5e7e001992d2b50c66d842e8b0674dd1977bdaa5b335770","target":"graph","created_at":"2026-07-05T11:18:46Z","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/2506.08187/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We establish a connection between a sharp double-sided Harnack bound for positive solutions of a fractional heat equation and the circular geometry in higher dimensions. The present work extends and generalizes the results obtained in the preceding paper.","authors_text":"Mateusz Dembny, Miko{\\l}aj Sier\\.z\\k{e}ga","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-06-09T19:52:43Z","title":"Geometric aspects of the Harnack Inequality for a nonlocal heat equation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.08187","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":"sha256:5b02d42260c6741b3c602a03a84c89e2ab295eec0b30082e04f68761516492dd","target":"record","created_at":"2026-07-05T11:18:46Z","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":"50072a95c1821e227ff5ef47fdf8604ea6d23ea5392df53506307d4871ccfe96","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-06-09T19:52:43Z","title_canon_sha256":"457ea712c5bec771cc82c8f15ed647d3818d7cc0d46af06d2af9107d472837be"},"schema_version":"1.0","source":{"id":"2506.08187","kind":"arxiv","version":1}},"canonical_sha256":"68530de41c25e0ce2defd71f182c7cc165a877a1a263349f771d56c0815a2b5d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"68530de41c25e0ce2defd71f182c7cc165a877a1a263349f771d56c0815a2b5d","first_computed_at":"2026-07-05T11:18:46.021408Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:18:46.021408Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"grmgwjTrqeBqMAEw6BphqXKIpgcV4JCFxgt0TwJMiwr+Gw7hM7mSZWCybsUHAvFU+JNde7YfcxbZexi107y6Dg==","signature_status":"signed_v1","signed_at":"2026-07-05T11:18:46.021827Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.08187","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5b02d42260c6741b3c602a03a84c89e2ab295eec0b30082e04f68761516492dd","sha256:431aa15eed5f808bc5e7e001992d2b50c66d842e8b0674dd1977bdaa5b335770"],"state_sha256":"28cef2576ca10826f258194d5cff8c3c543d2f197af7dcbfbb4ad8715ec703c0"}