{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:7UEZCPRRZ72Y5TFBJR223KHD3D","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":"7d6859c591cca7a4d07914ccaf66966ee8ee05b3f15003a56a54b8befe075db6","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.SC","submitted_at":"2026-02-12T04:48:27Z","title_canon_sha256":"90d93d1c29de6e6e13386af09a453d223ba64e437fd88a2432de5f133c53ce0b"},"schema_version":"1.0","source":{"id":"2603.00073","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2603.00073","created_at":"2026-05-17T23:38:59Z"},{"alias_kind":"arxiv_version","alias_value":"2603.00073v2","created_at":"2026-05-17T23:38:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2603.00073","created_at":"2026-05-17T23:38:59Z"},{"alias_kind":"pith_short_12","alias_value":"7UEZCPRRZ72Y","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"7UEZCPRRZ72Y5TFB","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"7UEZCPRR","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:8a3ec56883023c6b4c4da6c2dacbbdaf2a447b22eda260d15f1be77834fcd66d","target":"graph","created_at":"2026-05-17T23:38:59Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"necessary and sufficient conditions for the positivity of a quartic polynomial are derived through a separation method. Based on these conditions, more concise analytic expressions for the positivity of the Gram-Charlier density are proposed."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The separation method fully captures all cases of positivity for the specific quartic polynomials that arise from the Gram-Charlier expansion, without missing boundary cases or requiring additional restrictions."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"A separation method for quartic positivity produces more concise analytic conditions for the valid region of Gram-Charlier densities."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"A separation method supplies necessary and sufficient conditions for a quartic polynomial to stay positive everywhere, which then yields simpler analytic expressions for the valid region of Gram-Charlier densities."}],"snapshot_sha256":"ae0583fd61bb77b7077cee21005adf72f03aae0b546bce6ae81bc579b579a0d5"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"The positivity of the Gram-Charlier probability density function has been a subject of extensive study for decades. Since Barton and Dennis (1952) introduced numerical positivity conditions, no analytic closed-form expression was available until Kwon (2019, 2022) proposed analytic solutions for the valid region of Gram-Charlier densities. Despite the significance of the analytical solutions, the expressions remain algebraically complex. As these conditions for the Gram-Charlier densities are determined by a quartic polynomial, it is essential to investigate its positivity. In this work, necess","authors_text":"ByoungSeon Choi, Jung Chan Lee, Taehun Kim","cross_cats":[],"headline":"A separation method supplies necessary and sufficient conditions for a quartic polynomial to stay positive everywhere, which then yields simpler analytic expressions for the valid region of Gram-Charlier densities.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.SC","submitted_at":"2026-02-12T04:48:27Z","title":"A Separation Method for Quartic Positivity and the Valid Region of Gram-Charlier densities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2603.00073","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-16T02:12:07.394893Z","id":"3e91eccc-b5a4-4dd6-8680-db48dbcb4fa6","model_set":{"reader":"grok-4.3"},"one_line_summary":"A separation method for quartic positivity produces more concise analytic conditions for the valid region of Gram-Charlier densities.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"A separation method supplies necessary and sufficient conditions for a quartic polynomial to stay positive everywhere, which then yields simpler analytic expressions for the valid region of Gram-Charlier densities.","strongest_claim":"necessary and sufficient conditions for the positivity of a quartic polynomial are derived through a separation method. Based on these conditions, more concise analytic expressions for the positivity of the Gram-Charlier density are proposed.","weakest_assumption":"The separation method fully captures all cases of positivity for the specific quartic polynomials that arise from the Gram-Charlier expansion, without missing boundary cases or requiring additional restrictions."}},"verdict_id":"3e91eccc-b5a4-4dd6-8680-db48dbcb4fa6"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:53e1792e77ff0c611a34667da1d25db8d4f890438f59a32a750bded8e49f89e7","target":"record","created_at":"2026-05-17T23:38:59Z","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":"7d6859c591cca7a4d07914ccaf66966ee8ee05b3f15003a56a54b8befe075db6","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.SC","submitted_at":"2026-02-12T04:48:27Z","title_canon_sha256":"90d93d1c29de6e6e13386af09a453d223ba64e437fd88a2432de5f133c53ce0b"},"schema_version":"1.0","source":{"id":"2603.00073","kind":"arxiv","version":2}},"canonical_sha256":"fd09913e31cff58ecca14c75ada8e3d8ebbf849fafb03c4be4a8668f04208ea8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fd09913e31cff58ecca14c75ada8e3d8ebbf849fafb03c4be4a8668f04208ea8","first_computed_at":"2026-05-17T23:38:59.844062Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:38:59.844062Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"W+BwokXpSwao49aix7hOi2f7QSel2zBMYBdIlDthJcNGzFjJG9A2SsWYEP5hCCU/g7Mk4gFLHIpabG/3nDyTBw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:38:59.844750Z","signed_message":"canonical_sha256_bytes"},"source_id":"2603.00073","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:53e1792e77ff0c611a34667da1d25db8d4f890438f59a32a750bded8e49f89e7","sha256:8a3ec56883023c6b4c4da6c2dacbbdaf2a447b22eda260d15f1be77834fcd66d"],"state_sha256":"6430dc89d74231e419ebeff17ee7c099a79cafd6b15b3c954b1bd7d863ac74f2"}