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We prove that the spectrum of its $(r - 1)$-dimensional up-Laplacian is majorized by the conjugate degree sequence of its $(r - 1)$-dimensional faces: \\[ {\\mathbf{\\lambda}}_{r-1}(K) \\preccurlyeq {\\mathbf d}_{r-1}^\\top(K). \\] We also establish a Brouwer-type inequality: for every integer $\\ell \\geq 1$, \\[ \\sum_{i = 1}^{\\ell}\\lambda_{r-1,i}(K) \\leq \\frac{r + 1}{2}f_r(K) +\n\\frac{f_{r - 2}(K)}{r} \\binom{\\ell + 1}{2}, \\] where $\\lambda_{r-1,i}(K)$ denotes the $i$-th largest eigenvalue in the spectrum ${\\mathbf{\\lambda}}_{r-1}(K)$, and $f_t(K)$ denot"},"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":"2607.20910","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-23T04:20:41Z","cross_cats_sorted":[],"title_canon_sha256":"8675315a0a74e05ad4546ecad985fe1f9e957f3fa57af447d0f9463cc0ee4b4e","abstract_canon_sha256":"d97a4b35f11312a9441e548ef981b3e7f521f09d4f95868d551268dea743a2fb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-24T01:23:40.043105Z","signature_b64":"btfHHvnRLHnF98ttTC19e9sGnMdJg3qw2MD/th0yv6lCRJbi7B0Yv1AbxFjaru4HJbRGgZJuZYPW503L2/vECg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"033aab44e12ba5a0097da5ef2ef5db5dc2b5578d45adb0706d853ee8756ed560","last_reissued_at":"2026-07-24T01:23:40.042161Z","signature_status":"signed_v1","first_computed_at":"2026-07-24T01:23:40.042161Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Degree Majorization and Laplacian Eigenvalue Sums for Simplicial Complexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Huan-Zhi Zhang, Yi-Min Song, Yi-Zheng Fan","submitted_at":"2026-07-23T04:20:41Z","abstract_excerpt":"Let $K$ be an $r$-dimensional simplicial complex. We prove that the spectrum of its $(r - 1)$-dimensional up-Laplacian is majorized by the conjugate degree sequence of its $(r - 1)$-dimensional faces: \\[ {\\mathbf{\\lambda}}_{r-1}(K) \\preccurlyeq {\\mathbf d}_{r-1}^\\top(K). \\] We also establish a Brouwer-type inequality: for every integer $\\ell \\geq 1$, \\[ \\sum_{i = 1}^{\\ell}\\lambda_{r-1,i}(K) \\leq \\frac{r + 1}{2}f_r(K) +\n\\frac{f_{r - 2}(K)}{r} \\binom{\\ell + 1}{2}, \\] where $\\lambda_{r-1,i}(K)$ denotes the $i$-th largest eigenvalue in the spectrum ${\\mathbf{\\lambda}}_{r-1}(K)$, and $f_t(K)$ denot"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.20910","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/2607.20910/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":"2607.20910","created_at":"2026-07-24T01:23:40.042617+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.20910v1","created_at":"2026-07-24T01:23:40.042617+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.20910","created_at":"2026-07-24T01:23:40.042617+00:00"},{"alias_kind":"pith_short_12","alias_value":"AM5KWRHBFOS2","created_at":"2026-07-24T01:23:40.042617+00:00"},{"alias_kind":"pith_short_16","alias_value":"AM5KWRHBFOS2ACL5","created_at":"2026-07-24T01:23:40.042617+00:00"},{"alias_kind":"pith_short_8","alias_value":"AM5KWRHB","created_at":"2026-07-24T01:23:40.042617+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/AM5KWRHBFOS2ACL5UXXS55O3LX","json":"https://pith.science/pith/AM5KWRHBFOS2ACL5UXXS55O3LX.json","graph_json":"https://pith.science/api/pith-number/AM5KWRHBFOS2ACL5UXXS55O3LX/graph.json","events_json":"https://pith.science/api/pith-number/AM5KWRHBFOS2ACL5UXXS55O3LX/events.json","paper":"https://pith.science/paper/AM5KWRHB"},"agent_actions":{"view_html":"https://pith.science/pith/AM5KWRHBFOS2ACL5UXXS55O3LX","download_json":"https://pith.science/pith/AM5KWRHBFOS2ACL5UXXS55O3LX.json","view_paper":"https://pith.science/paper/AM5KWRHB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.20910&json=true","fetch_graph":"https://pith.science/api/pith-number/AM5KWRHBFOS2ACL5UXXS55O3LX/graph.json","fetch_events":"https://pith.science/api/pith-number/AM5KWRHBFOS2ACL5UXXS55O3LX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AM5KWRHBFOS2ACL5UXXS55O3LX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AM5KWRHBFOS2ACL5UXXS55O3LX/action/storage_attestation","attest_author":"https://pith.science/pith/AM5KWRHBFOS2ACL5UXXS55O3LX/action/author_attestation","sign_citation":"https://pith.science/pith/AM5KWRHBFOS2ACL5UXXS55O3LX/action/citation_signature","submit_replication":"https://pith.science/pith/AM5KWRHBFOS2ACL5UXXS55O3LX/action/replication_record"}},"created_at":"2026-07-24T01:23:40.042617+00:00","updated_at":"2026-07-24T01:23:40.042617+00:00"}