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For $\\sigma\\in X$, let $\\text{lk}(X,\\sigma)$ be the link of $\\sigma$ in $X$. For a matrix $M$, we denote by $\\text{Spec}(M)$ the multi-set containing all the eigenvalues of $M$. We show that, for every $0\\le \\ell<k \\le d$, \\[\n  \\text{dim}(\\tilde{H}_k(X;\\mathbb{R}))\\le \\sum_{\\eta\\"},"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":"2508.17279","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-08-24T10:00:57Z","cross_cats_sorted":[],"title_canon_sha256":"1ff072bdfaedd03fd0e2594a44bd933f1820f625c9688fa1d6ae94cbc2e2ed65","abstract_canon_sha256":"8e2d9bed86bb397c7d57f0b32a774b6832f183116605ce7e959e418ef8810456"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:58:39.043688Z","signature_b64":"bAglk4iTzeoaJjeL8PCz7YQCcOglmh6zhWETOzOKs7UhPdFVhT3FAV4YzJaLc32FVlZxvogCsBJMoV+3iSrHAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e1c45bfd9211e1218c7f9d78b80c313dd626566605e657e1817ba4503923afdc","last_reissued_at":"2026-07-05T11:58:39.043230Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:58:39.043230Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An eigenvalue interlacing approach to Garland's method","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alan Lew","submitted_at":"2025-08-24T10:00:57Z","abstract_excerpt":"Let $X$ be a pure $d$-dimensional simplicial complex. For $0\\le k\\le d$, let $X(k)$ be the set of $k$-dimensional faces of $X$, let $\\tilde{L}_k(X)$ be the $k$-dimensional weighted total Laplacian operator on $X$, and let $\\tilde{H}_k(X;\\mathbb{R})$ be its $k$-dimensional reduced homology group with real coefficients. For $\\sigma\\in X$, let $\\text{lk}(X,\\sigma)$ be the link of $\\sigma$ in $X$. For a matrix $M$, we denote by $\\text{Spec}(M)$ the multi-set containing all the eigenvalues of $M$. We show that, for every $0\\le \\ell<k \\le d$, \\[\n  \\text{dim}(\\tilde{H}_k(X;\\mathbb{R}))\\le \\sum_{\\eta\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.17279","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/2508.17279/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":"2508.17279","created_at":"2026-07-05T11:58:39.043289+00:00"},{"alias_kind":"arxiv_version","alias_value":"2508.17279v1","created_at":"2026-07-05T11:58:39.043289+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2508.17279","created_at":"2026-07-05T11:58:39.043289+00:00"},{"alias_kind":"pith_short_12","alias_value":"4HCFX7MSCHQS","created_at":"2026-07-05T11:58:39.043289+00:00"},{"alias_kind":"pith_short_16","alias_value":"4HCFX7MSCHQSDDD7","created_at":"2026-07-05T11:58:39.043289+00:00"},{"alias_kind":"pith_short_8","alias_value":"4HCFX7MS","created_at":"2026-07-05T11:58:39.043289+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2605.21069","citing_title":"The complex property of the boundary operator on simplicial complexes","ref_index":12,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/4HCFX7MSCHQSDDD7TV4LQDBRHX","json":"https://pith.science/pith/4HCFX7MSCHQSDDD7TV4LQDBRHX.json","graph_json":"https://pith.science/api/pith-number/4HCFX7MSCHQSDDD7TV4LQDBRHX/graph.json","events_json":"https://pith.science/api/pith-number/4HCFX7MSCHQSDDD7TV4LQDBRHX/events.json","paper":"https://pith.science/paper/4HCFX7MS"},"agent_actions":{"view_html":"https://pith.science/pith/4HCFX7MSCHQSDDD7TV4LQDBRHX","download_json":"https://pith.science/pith/4HCFX7MSCHQSDDD7TV4LQDBRHX.json","view_paper":"https://pith.science/paper/4HCFX7MS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2508.17279&json=true","fetch_graph":"https://pith.science/api/pith-number/4HCFX7MSCHQSDDD7TV4LQDBRHX/graph.json","fetch_events":"https://pith.science/api/pith-number/4HCFX7MSCHQSDDD7TV4LQDBRHX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4HCFX7MSCHQSDDD7TV4LQDBRHX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4HCFX7MSCHQSDDD7TV4LQDBRHX/action/storage_attestation","attest_author":"https://pith.science/pith/4HCFX7MSCHQSDDD7TV4LQDBRHX/action/author_attestation","sign_citation":"https://pith.science/pith/4HCFX7MSCHQSDDD7TV4LQDBRHX/action/citation_signature","submit_replication":"https://pith.science/pith/4HCFX7MSCHQSDDD7TV4LQDBRHX/action/replication_record"}},"created_at":"2026-07-05T11:58:39.043289+00:00","updated_at":"2026-07-05T11:58:39.043289+00:00"}