{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:N3HQWNEU6POODZ33WLCOYJ6HGU","short_pith_number":"pith:N3HQWNEU","canonical_record":{"source":{"id":"2607.15941","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-17T13:24:33Z","cross_cats_sorted":[],"title_canon_sha256":"e46e61d9c44b793e125f4395ae6f4a7a45eda8236e5cf953b8bbe00fa4d42659","abstract_canon_sha256":"4b58591bbc827c4ebed76f9465262193043496b403a8393887ae7cbcb4c8cec2"},"schema_version":"1.0"},"canonical_sha256":"6ecf0b3494f3dce1e77bb2c4ec27c735356ab6250b3ee43d9681f08caaf1c1d3","source":{"kind":"arxiv","id":"2607.15941","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.15941","created_at":"2026-07-20T01:19:18Z"},{"alias_kind":"arxiv_version","alias_value":"2607.15941v1","created_at":"2026-07-20T01:19:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.15941","created_at":"2026-07-20T01:19:18Z"},{"alias_kind":"pith_short_12","alias_value":"N3HQWNEU6POO","created_at":"2026-07-20T01:19:18Z"},{"alias_kind":"pith_short_16","alias_value":"N3HQWNEU6POODZ33","created_at":"2026-07-20T01:19:18Z"},{"alias_kind":"pith_short_8","alias_value":"N3HQWNEU","created_at":"2026-07-20T01:19:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:N3HQWNEU6POODZ33WLCOYJ6HGU","target":"record","payload":{"canonical_record":{"source":{"id":"2607.15941","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-17T13:24:33Z","cross_cats_sorted":[],"title_canon_sha256":"e46e61d9c44b793e125f4395ae6f4a7a45eda8236e5cf953b8bbe00fa4d42659","abstract_canon_sha256":"4b58591bbc827c4ebed76f9465262193043496b403a8393887ae7cbcb4c8cec2"},"schema_version":"1.0"},"canonical_sha256":"6ecf0b3494f3dce1e77bb2c4ec27c735356ab6250b3ee43d9681f08caaf1c1d3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-20T01:19:18.732858Z","signature_b64":"W92iwQDRogaYGDZY0Vu+ZedzE5Y2yhJmlPZ/y1WvdX4K/FVL4LqaByCENz+EkBLzSxk+qfiqnhEN6XVgSd+MAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6ecf0b3494f3dce1e77bb2c4ec27c735356ab6250b3ee43d9681f08caaf1c1d3","last_reissued_at":"2026-07-20T01:19:18.731933Z","signature_status":"signed_v1","first_computed_at":"2026-07-20T01:19:18.731933Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.15941","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-20T01:19:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"a7cMpT5UrhIMYRvh2nKdVYG9i7LJE59RFOxW6v5WtFDPMxSuy4ofJtucxq/uuAIERISzwNRuGr1aBR+AJGnQAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T22:37:51.610110Z"},"content_sha256":"3cba3d3543376c047d80cc3495fcf24f13cb264f9f7fff78c005eaf421297614","schema_version":"1.0","event_id":"sha256:3cba3d3543376c047d80cc3495fcf24f13cb264f9f7fff78c005eaf421297614"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:N3HQWNEU6POODZ33WLCOYJ6HGU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Generalized Nordhaus--Gaddum Inequalities for Eigenvalues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aiden Williams, Carter Antley, George Brooks, Ian Gonzalez, Joseph Aulenbacher, Linyuan Lu, Luke Hawranick, Sahil Agarwal, William Linz","submitted_at":"2026-07-17T13:24:33Z","abstract_excerpt":"For a graph $G$, let $\n\\lambda_1(G)\\ge \\lambda_2(G)\\ge \\cdots \\ge \\lambda_n(G)$ denote the adjacency eigenvalues of $G$. We investigate the asymptotic maximum of \\[ \\lambda_i(G)+\\lambda_j(\\overline G) \\] for fixed $i$ and $j$. We prove general bounds on $\\lambda_i(G) + \\lambda_{j}(\\overline{G})$ for all pairs $(i, j)$ and also give general bounds on the related problem of minimizing $\\lambda_{n-i+1}(G) + \\lambda_{n-j+1}(\\overline{G})$ for fixed $i$ and $j$. We prove that for all looped graphs $G$ on $n$ vertices, \\[\\lambda_1(G) + \\lambda_2(\\overline{G}) \\le \\frac87 n. \\] Our method also gives "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.15941","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.15941/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-20T01:19:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9wAL0jTmqVrgi2jFxYBEHwxfUL3Eg4B2lOJP8B9ZzPJ4w/ZiRFY7lGsN8/5gJJYMo9Ain9KBae2coowX+XptCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T22:37:51.610628Z"},"content_sha256":"591cbc575c5afdbaa620e72a9d51c3d4adf9647bd832b44546b61f9e3e9646e7","schema_version":"1.0","event_id":"sha256:591cbc575c5afdbaa620e72a9d51c3d4adf9647bd832b44546b61f9e3e9646e7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/N3HQWNEU6POODZ33WLCOYJ6HGU/bundle.json","state_url":"https://pith.science/pith/N3HQWNEU6POODZ33WLCOYJ6HGU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/N3HQWNEU6POODZ33WLCOYJ6HGU/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-18T22:37:51Z","links":{"resolver":"https://pith.science/pith/N3HQWNEU6POODZ33WLCOYJ6HGU","bundle":"https://pith.science/pith/N3HQWNEU6POODZ33WLCOYJ6HGU/bundle.json","state":"https://pith.science/pith/N3HQWNEU6POODZ33WLCOYJ6HGU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/N3HQWNEU6POODZ33WLCOYJ6HGU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:N3HQWNEU6POODZ33WLCOYJ6HGU","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":"4b58591bbc827c4ebed76f9465262193043496b403a8393887ae7cbcb4c8cec2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-17T13:24:33Z","title_canon_sha256":"e46e61d9c44b793e125f4395ae6f4a7a45eda8236e5cf953b8bbe00fa4d42659"},"schema_version":"1.0","source":{"id":"2607.15941","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.15941","created_at":"2026-07-20T01:19:18Z"},{"alias_kind":"arxiv_version","alias_value":"2607.15941v1","created_at":"2026-07-20T01:19:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.15941","created_at":"2026-07-20T01:19:18Z"},{"alias_kind":"pith_short_12","alias_value":"N3HQWNEU6POO","created_at":"2026-07-20T01:19:18Z"},{"alias_kind":"pith_short_16","alias_value":"N3HQWNEU6POODZ33","created_at":"2026-07-20T01:19:18Z"},{"alias_kind":"pith_short_8","alias_value":"N3HQWNEU","created_at":"2026-07-20T01:19:18Z"}],"graph_snapshots":[{"event_id":"sha256:591cbc575c5afdbaa620e72a9d51c3d4adf9647bd832b44546b61f9e3e9646e7","target":"graph","created_at":"2026-07-20T01:19:18Z","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/2607.15941/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For a graph $G$, let $\n\\lambda_1(G)\\ge \\lambda_2(G)\\ge \\cdots \\ge \\lambda_n(G)$ denote the adjacency eigenvalues of $G$. We investigate the asymptotic maximum of \\[ \\lambda_i(G)+\\lambda_j(\\overline G) \\] for fixed $i$ and $j$. We prove general bounds on $\\lambda_i(G) + \\lambda_{j}(\\overline{G})$ for all pairs $(i, j)$ and also give general bounds on the related problem of minimizing $\\lambda_{n-i+1}(G) + \\lambda_{n-j+1}(\\overline{G})$ for fixed $i$ and $j$. We prove that for all looped graphs $G$ on $n$ vertices, \\[\\lambda_1(G) + \\lambda_2(\\overline{G}) \\le \\frac87 n. \\] Our method also gives ","authors_text":"Aiden Williams, Carter Antley, George Brooks, Ian Gonzalez, Joseph Aulenbacher, Linyuan Lu, Luke Hawranick, Sahil Agarwal, William Linz","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-17T13:24:33Z","title":"Generalized Nordhaus--Gaddum Inequalities for Eigenvalues"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.15941","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":"sha256:3cba3d3543376c047d80cc3495fcf24f13cb264f9f7fff78c005eaf421297614","target":"record","created_at":"2026-07-20T01:19:18Z","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":"4b58591bbc827c4ebed76f9465262193043496b403a8393887ae7cbcb4c8cec2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-17T13:24:33Z","title_canon_sha256":"e46e61d9c44b793e125f4395ae6f4a7a45eda8236e5cf953b8bbe00fa4d42659"},"schema_version":"1.0","source":{"id":"2607.15941","kind":"arxiv","version":1}},"canonical_sha256":"6ecf0b3494f3dce1e77bb2c4ec27c735356ab6250b3ee43d9681f08caaf1c1d3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6ecf0b3494f3dce1e77bb2c4ec27c735356ab6250b3ee43d9681f08caaf1c1d3","first_computed_at":"2026-07-20T01:19:18.731933Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-20T01:19:18.731933Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"W92iwQDRogaYGDZY0Vu+ZedzE5Y2yhJmlPZ/y1WvdX4K/FVL4LqaByCENz+EkBLzSxk+qfiqnhEN6XVgSd+MAw==","signature_status":"signed_v1","signed_at":"2026-07-20T01:19:18.732858Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.15941","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3cba3d3543376c047d80cc3495fcf24f13cb264f9f7fff78c005eaf421297614","sha256:591cbc575c5afdbaa620e72a9d51c3d4adf9647bd832b44546b61f9e3e9646e7"],"state_sha256":"49a73aca5cbf608c75c73ebfe9002d941322cd9f1f41c95f095427bd224a028d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9tCobQW/0WM7sNPOCSFMhlS9PGclmhVXFYClU9iNsfgwJ3u6Ppy53KEJ3AS/4SlAFIAYRyDOvF/WZGer3+u+Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-18T22:37:51.614168Z","bundle_sha256":"7b460773353c360fca0d9058678bb631ceab38f2e0a2538918d3d6b26b6bd82a"}}