{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:F73PCZRWAPNNA74OF2NVNYZ32F","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":"7a24234fa41512c2966be038468c6da6495d343bae0ffe125ef5ca9a79f2261b","cross_cats_sorted":["math.CO","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2025-08-28T14:14:08Z","title_canon_sha256":"3cd5e7462e8004ee4e9c3d7dd604b07f095f7b2d485c03429485070cf742ec7c"},"schema_version":"1.0","source":{"id":"2508.20815","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2508.20815","created_at":"2026-07-05T12:01:11Z"},{"alias_kind":"arxiv_version","alias_value":"2508.20815v1","created_at":"2026-07-05T12:01:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2508.20815","created_at":"2026-07-05T12:01:11Z"},{"alias_kind":"pith_short_12","alias_value":"F73PCZRWAPNN","created_at":"2026-07-05T12:01:11Z"},{"alias_kind":"pith_short_16","alias_value":"F73PCZRWAPNNA74O","created_at":"2026-07-05T12:01:11Z"},{"alias_kind":"pith_short_8","alias_value":"F73PCZRW","created_at":"2026-07-05T12:01:11Z"}],"graph_snapshots":[{"event_id":"sha256:c49dee7349eae4fcc39cc89cf90392c1d89ebfddeb3bf1a70f2ad1a4d947d7e8","target":"graph","created_at":"2026-07-05T12:01:11Z","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/2508.20815/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Under mild assumptions, we show that a connected weighted graph $G$ with lower Ricci curvature bound $K>0$ in the sense of Bakry-\\'Emery and the $d$-th non-zero Laplacian eigenvalue $\\lambda_d$ close to $K$, with $d$ being the maximal combinatorial vertex degree of $G$, has an underlying combinatorial structure of the $d$-dimensional hypercube graph. Moreover, such a graph $G$ is close in terms of Frobenius distance to a properly weighted hypercube graph. Furthermore, we establish their closeness in terms of eigenfunctions. Our results can be viewed as discrete analogies of the almost rigidity","authors_text":"Chiyu Zhou, Shiping Liu","cross_cats":["math.CO","math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2025-08-28T14:14:08Z","title":"Quantitative Obata's theorem in discrete setting"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.20815","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:aa46d2c810cb16ed630bb81e40e5c53a6db3c43299af82fc63c5e7bee5eb107d","target":"record","created_at":"2026-07-05T12:01:11Z","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":"7a24234fa41512c2966be038468c6da6495d343bae0ffe125ef5ca9a79f2261b","cross_cats_sorted":["math.CO","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2025-08-28T14:14:08Z","title_canon_sha256":"3cd5e7462e8004ee4e9c3d7dd604b07f095f7b2d485c03429485070cf742ec7c"},"schema_version":"1.0","source":{"id":"2508.20815","kind":"arxiv","version":1}},"canonical_sha256":"2ff6f1663603dad07f8e2e9b56e33bd14aa1d5ca8993675fe7e205040f5c419a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2ff6f1663603dad07f8e2e9b56e33bd14aa1d5ca8993675fe7e205040f5c419a","first_computed_at":"2026-07-05T12:01:11.531593Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T12:01:11.531593Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1xC6zXI37UNA0VeOb+sB8b1C/lHTX1n0X56VZFBid4O628TncvUEkHRfK2M0f3BrDJvqw1FbiWLRaHQ8T8/dAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T12:01:11.532002Z","signed_message":"canonical_sha256_bytes"},"source_id":"2508.20815","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:aa46d2c810cb16ed630bb81e40e5c53a6db3c43299af82fc63c5e7bee5eb107d","sha256:c49dee7349eae4fcc39cc89cf90392c1d89ebfddeb3bf1a70f2ad1a4d947d7e8"],"state_sha256":"d8d9d1d193c1d8e21bde6263fcafd38d16157a310b004ea6087091d24256969c"}