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P\\'olya conjectured in 1954 that each Dirichlet eigenvalue $\\lambda_k(\\Omega)$ is greater than $w_k(\\Omega)$, while each Neumann eigenvalue $\\mu_k(\\Omega)$ is no more than $w_k(\\Omega)$. In this paper we prove P\\'olya's conjecture for thin products, i.e. domains of the form $(a\\Omega_1) \\times \\Omega_2$, where $\\Omega_"},"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":"2402.12093","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.SP","submitted_at":"2024-02-19T12:16:15Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"6c49aae2d14f365373e3e8577876a7b7a01ccd3726b58f20a4e43f8e237c3a4c","abstract_canon_sha256":"9ddb56930ecfdcf88cccd53826d6954456c9c2e929fc74af507afe945985686f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:33:57.468241Z","signature_b64":"ZLcstLkzQpkVjYJIU67wgmgEl1Ll4NlNWN/akuk3bAszeb6NXGsWtnWpuft9zFLtabtalIKWVT3Db1BrdCxyCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4908801bd29c280eab56e126919848a5f78a9d8b757b110272baaab14cfb7fee","last_reissued_at":"2026-07-05T11:33:57.467769Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:33:57.467769Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"P\\'olya's conjecture for thin products","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.SP","authors_text":"Xiang He, Zuoqin Wang","submitted_at":"2024-02-19T12:16:15Z","abstract_excerpt":"Let $\\Omega \\subset \\mathbb R^d$ be a bounded Euclidean domain. According to the famous Weyl law, both its Dirichlet eigenvalue $\\lambda_k(\\Omega)$ and its Neumann eigenvalue $\\mu_k(\\Omega)$ have the same leading asymptotics $w_k(\\Omega)=C(d,\\Omega)k^{2/d}$ as $k \\to \\infty$. G. P\\'olya conjectured in 1954 that each Dirichlet eigenvalue $\\lambda_k(\\Omega)$ is greater than $w_k(\\Omega)$, while each Neumann eigenvalue $\\mu_k(\\Omega)$ is no more than $w_k(\\Omega)$. In this paper we prove P\\'olya's conjecture for thin products, i.e. domains of the form $(a\\Omega_1) \\times \\Omega_2$, where $\\Omega_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.12093","kind":"arxiv","version":3},"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/2402.12093/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":"2402.12093","created_at":"2026-07-05T11:33:57.467821+00:00"},{"alias_kind":"arxiv_version","alias_value":"2402.12093v3","created_at":"2026-07-05T11:33:57.467821+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2402.12093","created_at":"2026-07-05T11:33:57.467821+00:00"},{"alias_kind":"pith_short_12","alias_value":"JEEIAG6STQUA","created_at":"2026-07-05T11:33:57.467821+00:00"},{"alias_kind":"pith_short_16","alias_value":"JEEIAG6STQUA5K2W","created_at":"2026-07-05T11:33:57.467821+00:00"},{"alias_kind":"pith_short_8","alias_value":"JEEIAG6S","created_at":"2026-07-05T11:33:57.467821+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":2,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.11659","citing_title":"Krahn--Szeg\\H{o} type inequalities and nodal domain methods on graphs","ref_index":23,"is_internal_anchor":false},{"citing_arxiv_id":"2604.20367","citing_title":"Improved lower bounds for Dirichlet eigenvalues of the Laplacian and poly-Laplacian on bounded Euclidean domains","ref_index":13,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/JEEIAG6STQUA5K2W4ETJDGCIUX","json":"https://pith.science/pith/JEEIAG6STQUA5K2W4ETJDGCIUX.json","graph_json":"https://pith.science/api/pith-number/JEEIAG6STQUA5K2W4ETJDGCIUX/graph.json","events_json":"https://pith.science/api/pith-number/JEEIAG6STQUA5K2W4ETJDGCIUX/events.json","paper":"https://pith.science/paper/JEEIAG6S"},"agent_actions":{"view_html":"https://pith.science/pith/JEEIAG6STQUA5K2W4ETJDGCIUX","download_json":"https://pith.science/pith/JEEIAG6STQUA5K2W4ETJDGCIUX.json","view_paper":"https://pith.science/paper/JEEIAG6S","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2402.12093&json=true","fetch_graph":"https://pith.science/api/pith-number/JEEIAG6STQUA5K2W4ETJDGCIUX/graph.json","fetch_events":"https://pith.science/api/pith-number/JEEIAG6STQUA5K2W4ETJDGCIUX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JEEIAG6STQUA5K2W4ETJDGCIUX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JEEIAG6STQUA5K2W4ETJDGCIUX/action/storage_attestation","attest_author":"https://pith.science/pith/JEEIAG6STQUA5K2W4ETJDGCIUX/action/author_attestation","sign_citation":"https://pith.science/pith/JEEIAG6STQUA5K2W4ETJDGCIUX/action/citation_signature","submit_replication":"https://pith.science/pith/JEEIAG6STQUA5K2W4ETJDGCIUX/action/replication_record"}},"created_at":"2026-07-05T11:33:57.467821+00:00","updated_at":"2026-07-05T11:33:57.467821+00:00"}