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Then $D_r(F)=\\sum_{v\\in V}\\min\\{d_F(v),r\\}$ is the \\(r\\)-th partial sum of the conjugate degree sequence of \\(F\\). The majorization assertion in the Duval--Reiner conjecture [Trans. Amer. Math. Soc., 2002] states that \\(s_r(F)\\le D_r(F)\\) for every \\(q\\)-uniform family \\(F\\) and every \\(r\\ge1\\). We disprove this assertion in two complementary senses: for every \\(r\\ge5\\), t"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2607.20051","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-22T11:49:13Z","cross_cats_sorted":[],"title_canon_sha256":"53b7e89aa8540d583c003ccf4dcf87d22e191d35d2ec8848fa3f04ff94fad733","abstract_canon_sha256":"0376fb2d643e78ea9b84213f9cf2a65c97ed9cbef47d75afab7c29560b7c552a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-23T01:24:57.427570Z","signature_b64":"8O2MRSWkmXLm8qbvwlrXDwxRBEWLCG3csQu61EZ9N3VUvtbZVkFx3WhlWtn0BkujwIzem1WRWEw1aa5plnU3Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3fbbe71cfc8e3dd5fa84a1e76dfabb61c8d3f65c37efcd404cc94de08cb1fb7d","last_reissued_at":"2026-07-23T01:24:57.426709Z","signature_status":"signed_v1","first_computed_at":"2026-07-23T01:24:57.426709Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Duval--Reiner Conjecture: Counterexamples and the Second Partial-Sum Inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jing Huang","submitted_at":"2026-07-22T11:49:13Z","abstract_excerpt":"Let \\(F\\subseteq\\binom{V}{q}\\) be a \\(q\\)-uniform family on a finite vertex set \\(V\\). Write \\(s_r(F)\\) for the sum of the \\(r\\) largest eigenvalues of its simplicial up-Laplacian and \\(d_F(v)\\) for the degree of \\(v\\in V\\). Then $D_r(F)=\\sum_{v\\in V}\\min\\{d_F(v),r\\}$ is the \\(r\\)-th partial sum of the conjugate degree sequence of \\(F\\). The majorization assertion in the Duval--Reiner conjecture [Trans. Amer. Math. Soc., 2002] states that \\(s_r(F)\\le D_r(F)\\) for every \\(q\\)-uniform family \\(F\\) and every \\(r\\ge1\\). We disprove this assertion in two complementary senses: for every \\(r\\ge5\\), t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.20051","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/2607.20051/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2607.20051","created_at":"2026-07-23T01:24:57.427147+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.20051v1","created_at":"2026-07-23T01:24:57.427147+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.20051","created_at":"2026-07-23T01:24:57.427147+00:00"},{"alias_kind":"pith_short_12","alias_value":"H656OHH4RY65","created_at":"2026-07-23T01:24:57.427147+00:00"},{"alias_kind":"pith_short_16","alias_value":"H656OHH4RY65L6UE","created_at":"2026-07-23T01:24:57.427147+00:00"},{"alias_kind":"pith_short_8","alias_value":"H656OHH4","created_at":"2026-07-23T01:24:57.427147+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/H656OHH4RY65L6UEUHTW36V3MH","json":"https://pith.science/pith/H656OHH4RY65L6UEUHTW36V3MH.json","graph_json":"https://pith.science/api/pith-number/H656OHH4RY65L6UEUHTW36V3MH/graph.json","events_json":"https://pith.science/api/pith-number/H656OHH4RY65L6UEUHTW36V3MH/events.json","paper":"https://pith.science/paper/H656OHH4"},"agent_actions":{"view_html":"https://pith.science/pith/H656OHH4RY65L6UEUHTW36V3MH","download_json":"https://pith.science/pith/H656OHH4RY65L6UEUHTW36V3MH.json","view_paper":"https://pith.science/paper/H656OHH4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.20051&json=true","fetch_graph":"https://pith.science/api/pith-number/H656OHH4RY65L6UEUHTW36V3MH/graph.json","fetch_events":"https://pith.science/api/pith-number/H656OHH4RY65L6UEUHTW36V3MH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/H656OHH4RY65L6UEUHTW36V3MH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/H656OHH4RY65L6UEUHTW36V3MH/action/storage_attestation","attest_author":"https://pith.science/pith/H656OHH4RY65L6UEUHTW36V3MH/action/author_attestation","sign_citation":"https://pith.science/pith/H656OHH4RY65L6UEUHTW36V3MH/action/citation_signature","submit_replication":"https://pith.science/pith/H656OHH4RY65L6UEUHTW36V3MH/action/replication_record"}},"created_at":"2026-07-23T01:24:57.427147+00:00","updated_at":"2026-07-23T01:24:57.427147+00:00"}