{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:RGWDAK5ROZFXDKW54Z2RVWBWGP","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":"36d641cdd72a9ac31ea00a898b61eedd6cf9dad335994b6897b3c0985856940a","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-18T10:16:58Z","title_canon_sha256":"3b8df7f3fafd2057bf5f82b21e037fcaa9dfd4bd01f8b29e435aeebace69d8fc"},"schema_version":"1.0","source":{"id":"2607.16746","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.16746","created_at":"2026-07-21T01:20:24Z"},{"alias_kind":"arxiv_version","alias_value":"2607.16746v1","created_at":"2026-07-21T01:20:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.16746","created_at":"2026-07-21T01:20:24Z"},{"alias_kind":"pith_short_12","alias_value":"RGWDAK5ROZFX","created_at":"2026-07-21T01:20:24Z"},{"alias_kind":"pith_short_16","alias_value":"RGWDAK5ROZFXDKW5","created_at":"2026-07-21T01:20:24Z"},{"alias_kind":"pith_short_8","alias_value":"RGWDAK5R","created_at":"2026-07-21T01:20:24Z"}],"graph_snapshots":[{"event_id":"sha256:27d1030fcded3842d848c5a6fa3339938074730f282af7c99266e6c05cc998ae","target":"graph","created_at":"2026-07-21T01:20:24Z","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/2607.16746/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we use the spectral surplus $\\lambda(G) - \\sqrt{m}$ to measure how far $G$ lies above the Nosal threshold, and prove the following edge-spectral supersaturation results for triangles and books.\n  (a) Every graph $G$ with $m\\ge 3$ edges and $\\lambda(G) \\ge 1 + \\sqrt{m-2}$ contains at least $m-2$ triangles, with equality if and only if $G = K_3 \\vee \\tfrac{m-3}{3} K_1$. This can be viewed as the third-layer supersaturation in the jump phenomenon, after the first layer $t(G) \\ge \\lfloor \\tfrac{1}{2}(\\sqrt{m}-1) \\rfloor$ proved by Ning and Zhai, and the second layer $t(G) \\ge \\tfrac","authors_text":"Hongzhang Chen, Quanyu Tang, Yongtao Li","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-18T10:16:58Z","title":"Supersaturation in Nosal graphs: Triangles and books"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.16746","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":"sha256:46842e17c6e98ac6c64bfd072abde73231f74deaa440bc73864c32fb7462f3e6","target":"record","created_at":"2026-07-21T01:20:24Z","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":"36d641cdd72a9ac31ea00a898b61eedd6cf9dad335994b6897b3c0985856940a","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-07-18T10:16:58Z","title_canon_sha256":"3b8df7f3fafd2057bf5f82b21e037fcaa9dfd4bd01f8b29e435aeebace69d8fc"},"schema_version":"1.0","source":{"id":"2607.16746","kind":"arxiv","version":1}},"canonical_sha256":"89ac302bb1764b71aadde6751ad83633ff24e923741fd10656026e9016d9a370","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"89ac302bb1764b71aadde6751ad83633ff24e923741fd10656026e9016d9a370","first_computed_at":"2026-07-21T01:20:24.893272Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-21T01:20:24.893272Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AzkqAJuyUjTyNx6sLMNMG1lAOY4cwplUaZ7BVITUbsaTHAW3xQt5+SN76pHLfz9YOvh/My8bX3rdtGjjmkRyAA==","signature_status":"signed_v1","signed_at":"2026-07-21T01:20:24.894172Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.16746","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:46842e17c6e98ac6c64bfd072abde73231f74deaa440bc73864c32fb7462f3e6","sha256:27d1030fcded3842d848c5a6fa3339938074730f282af7c99266e6c05cc998ae"],"state_sha256":"f7b8ba0c84748906c2c98d0b5893104c507e718a30f573e218cc80ab5886b48d"}