{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:4HCFX7MSCHQSDDD7TV4LQDBRHX","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":"8e2d9bed86bb397c7d57f0b32a774b6832f183116605ce7e959e418ef8810456","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-08-24T10:00:57Z","title_canon_sha256":"1ff072bdfaedd03fd0e2594a44bd933f1820f625c9688fa1d6ae94cbc2e2ed65"},"schema_version":"1.0","source":{"id":"2508.17279","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2508.17279","created_at":"2026-07-05T11:58:39Z"},{"alias_kind":"arxiv_version","alias_value":"2508.17279v1","created_at":"2026-07-05T11:58:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2508.17279","created_at":"2026-07-05T11:58:39Z"},{"alias_kind":"pith_short_12","alias_value":"4HCFX7MSCHQS","created_at":"2026-07-05T11:58:39Z"},{"alias_kind":"pith_short_16","alias_value":"4HCFX7MSCHQSDDD7","created_at":"2026-07-05T11:58:39Z"},{"alias_kind":"pith_short_8","alias_value":"4HCFX7MS","created_at":"2026-07-05T11:58:39Z"}],"graph_snapshots":[{"event_id":"sha256:6e591354f0e2d952f40ffdd913b9f2768f75bf69788d05d298fc8b39569cc327","target":"graph","created_at":"2026-07-05T11:58:39Z","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/2508.17279/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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\\","authors_text":"Alan Lew","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-08-24T10:00:57Z","title":"An eigenvalue interlacing approach to Garland's method"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.17279","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":"sha256:333b97536cf0ff3d7919a9e298d4f0ba3f183aa40b480d06722dc522b31e67a9","target":"record","created_at":"2026-07-05T11:58:39Z","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":"8e2d9bed86bb397c7d57f0b32a774b6832f183116605ce7e959e418ef8810456","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-08-24T10:00:57Z","title_canon_sha256":"1ff072bdfaedd03fd0e2594a44bd933f1820f625c9688fa1d6ae94cbc2e2ed65"},"schema_version":"1.0","source":{"id":"2508.17279","kind":"arxiv","version":1}},"canonical_sha256":"e1c45bfd9211e1218c7f9d78b80c313dd626566605e657e1817ba4503923afdc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e1c45bfd9211e1218c7f9d78b80c313dd626566605e657e1817ba4503923afdc","first_computed_at":"2026-07-05T11:58:39.043230Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:58:39.043230Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bAglk4iTzeoaJjeL8PCz7YQCcOglmh6zhWETOzOKs7UhPdFVhT3FAV4YzJaLc32FVlZxvogCsBJMoV+3iSrHAw==","signature_status":"signed_v1","signed_at":"2026-07-05T11:58:39.043688Z","signed_message":"canonical_sha256_bytes"},"source_id":"2508.17279","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:333b97536cf0ff3d7919a9e298d4f0ba3f183aa40b480d06722dc522b31e67a9","sha256:6e591354f0e2d952f40ffdd913b9f2768f75bf69788d05d298fc8b39569cc327"],"state_sha256":"aaa0ad941c01c2a360a3a8e70abb5b8f99f8db524fbee99c870ae28c4ce2a5cb"}