{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:TBFXXZTRJ7HTWQMEQFMBAJEING","merge_version":"pith-open-graph-merge-v1","event_count":7,"valid_event_count":7,"invalid_event_count":0,"equivocation_count":1,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"6fe20f095f5b91d2a6d5633e8e57a6b087db414fd840f1a29b637890643d0c63","cross_cats_sorted":["math.DG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-07-23T08:07:44Z","title_canon_sha256":"ceef766690f735aa38501b17725b9ddc7c09aa66bba713336c67485307d5a89b"},"schema_version":"1.0","source":{"id":"2607.21024","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.21024","created_at":"2026-07-24T01:23:47Z"},{"alias_kind":"arxiv_version","alias_value":"2607.21024v1","created_at":"2026-07-24T01:23:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.21024","created_at":"2026-07-24T01:23:47Z"},{"alias_kind":"pith_short_12","alias_value":"TBFXXZTRJ7HT","created_at":"2026-07-24T01:23:47Z"},{"alias_kind":"pith_short_16","alias_value":"TBFXXZTRJ7HTWQME","created_at":"2026-07-24T01:23:47Z"},{"alias_kind":"pith_short_8","alias_value":"TBFXXZTR","created_at":"2026-07-24T01:23:47Z"}],"graph_snapshots":[{"event_id":"sha256:dd90afb3ce0e1720e2301ab6c51b6a20f69b6834073ea5319cc8441bfcd7b7d3","target":"graph","created_at":"2026-07-24T01:23:47Z","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/2607.21024/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove Liouville rigidity theorems for translation-invariant real and complex Hessian equations in the viscosity sense, where the PDE is encoded by an admissible set $\\mathcal{A}$. The main structural notion is Liouville admissibility, a recursive geometric condition requiring each quotient set to be either boundary compatible or to fall into a terminal class. Our main theorem states that every bounded, globally $C^{0,\\alpha}$ entire viscosity solution of \\[ \\mathrm{Hess}_{\\mathbb F}u\\in\\partial\\mathcal{A} \\]\n  is constant if and only if $\\mathcal{A}$ is Liouville admissible; thus the Liouvi","authors_text":"Biao Ma, Hao Fang, Jinyang Wu","cross_cats":["math.DG"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-07-23T08:07:44Z","title":"Liouville Rigidity for Real and Complex Degenerate Hessian Equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.21024","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":"sha256:55f4b48fde4da0ee481836f1e3d9c8b101b3884d06a99e1ed4a373237122aaf4","target":"record","created_at":"2026-07-24T01:23:47Z","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":"6fe20f095f5b91d2a6d5633e8e57a6b087db414fd840f1a29b637890643d0c63","cross_cats_sorted":["math.DG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-07-23T08:07:44Z","title_canon_sha256":"ceef766690f735aa38501b17725b9ddc7c09aa66bba713336c67485307d5a89b"},"schema_version":"1.0","source":{"id":"2607.21024","kind":"arxiv","version":1}},"canonical_sha256":"984b7be6714fcf3b4184815810248869a86ee18301698d44f6ffb17ad1589f31","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"984b7be6714fcf3b4184815810248869a86ee18301698d44f6ffb17ad1589f31","first_computed_at":"2026-07-24T01:23:47.900354Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-24T01:23:47.900354Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"W9tZ3qW2agKBJz9yOmK+ghjkHrFqk5HN2hNvZzZyRbPRhzJaulGtd2uy9l42P+PkKrnRWBg8OsSnxPG2Lj2aBw==","signature_status":"signed_v1","signed_at":"2026-07-24T01:23:47.901173Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.21024","source_kind":"arxiv","source_version":1}}},"equivocations":[{"signer_id":"pith.science","event_type":"integrity_finding","target":"integrity","event_ids":["sha256:6822c5cfb05eb9532524dde6f2bfb7e3030aaa9deed4f47dbb2330098a706bb2","sha256:ad7561844df579a529a4991eb498a7fc2516483571c147d2eb8b8227524daa5c","sha256:e07af492ddf449ee076e6d1ec4c738dac1d0885b0c1b98450ce8c1230816110f","sha256:ed2a75bdb5b6fe23335e26cb82b5d2a75ab29fbf00288d99e3b212dfe96b0b16","sha256:f121fd20e3838c38925de86bed15fb193ed487f937cd7fe269460fe787958c32"]}],"invalid_events":[],"applied_event_ids":["sha256:55f4b48fde4da0ee481836f1e3d9c8b101b3884d06a99e1ed4a373237122aaf4","sha256:dd90afb3ce0e1720e2301ab6c51b6a20f69b6834073ea5319cc8441bfcd7b7d3"],"state_sha256":"c652904735418537634420a0f4f7f788c873b08d3b1560f3bb661bc8d2890fed"}