{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:JWPRRSL2ZL4K4SSL5SGTFX4MCX","short_pith_number":"pith:JWPRRSL2","canonical_record":{"source":{"id":"1411.1135","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.DG","submitted_at":"2014-11-05T03:06:46Z","cross_cats_sorted":["math.AP","math.SP"],"title_canon_sha256":"df35a48c918b134b27acc51d54ecfc2fd6986f16016d9cc66fa2b1924eac6cb7","abstract_canon_sha256":"9173f6718df50e0edcba27491a0080795345e912e06fa40db8ee57730b621589"},"schema_version":"1.0"},"canonical_sha256":"4d9f18c97acaf8ae4a4bec8d32df8c15cf532ef404b6110990ca85ab0a9b59a6","source":{"kind":"arxiv","id":"1411.1135","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1411.1135","created_at":"2026-05-18T02:38:08Z"},{"alias_kind":"arxiv_version","alias_value":"1411.1135v2","created_at":"2026-05-18T02:38:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.1135","created_at":"2026-05-18T02:38:08Z"},{"alias_kind":"pith_short_12","alias_value":"JWPRRSL2ZL4K","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_16","alias_value":"JWPRRSL2ZL4K4SSL","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_8","alias_value":"JWPRRSL2","created_at":"2026-05-18T12:28:35Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:JWPRRSL2ZL4K4SSL5SGTFX4MCX","target":"record","payload":{"canonical_record":{"source":{"id":"1411.1135","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.DG","submitted_at":"2014-11-05T03:06:46Z","cross_cats_sorted":["math.AP","math.SP"],"title_canon_sha256":"df35a48c918b134b27acc51d54ecfc2fd6986f16016d9cc66fa2b1924eac6cb7","abstract_canon_sha256":"9173f6718df50e0edcba27491a0080795345e912e06fa40db8ee57730b621589"},"schema_version":"1.0"},"canonical_sha256":"4d9f18c97acaf8ae4a4bec8d32df8c15cf532ef404b6110990ca85ab0a9b59a6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:38:08.767093Z","signature_b64":"DC5yNpgEaLV4i93oSD2fZhV0GWdhfrK0kRcaqIEfpY1CV0WXrP4i1fBxoR/KuSDYW+U7ORTWedvNutjMRPD/CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4d9f18c97acaf8ae4a4bec8d32df8c15cf532ef404b6110990ca85ab0a9b59a6","last_reissued_at":"2026-05-18T02:38:08.766456Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:38:08.766456Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1411.1135","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:38:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YwzcNXAjn9bp1sPMpOVFD5f4LvSMBuFXoPvvtjXMXiBl8S0OKR50658OpFReTHfDS/i2dnpaKT21GWdnD5+WDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T07:25:24.128409Z"},"content_sha256":"02ba3e8ddbbe3aa140d796727c757e88b17843a1549e0d2cd6184cfa2c257190","schema_version":"1.0","event_id":"sha256:02ba3e8ddbbe3aa140d796727c757e88b17843a1549e0d2cd6184cfa2c257190"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:JWPRRSL2ZL4K4SSL5SGTFX4MCX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Proof of the P\\'{o}lya conjecture","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":["math.AP","math.SP"],"primary_cat":"math.DG","authors_text":"Yue He","submitted_at":"2014-11-05T03:06:46Z","abstract_excerpt":"In this paper, we study lower bounds for higher eigenvalues of the Dirichlet eigenvalue problem of the Laplacian on a bounded domain $\\Omega$ in $\\mathbb{R}^n$. It is well known that the $k$-th Dirichlet eigenvalue $\\lambda_k$ obeys the Weyl asymptotic formula, that is,\n\\[\n\\lambda_k\\sim\\frac{4\\pi^2}{(\\omega_n\\mathrm{vol}\\Omega)^\\frac{2}{n}}k^\\frac{2}{n}\\qquad\\hbox{as}\\quad k\\rightarrow\\infty,\n\\]\nwhere $\\mathrm{vol}\\Omega$ is the volume of $\\Omega$. In view of the above formula, P\\'{o}lya conjectured that\n\\[\n\\lambda_k\\gs\\frac{4\\pi^2}{(\\omega_n\\mathrm{vol}\\Omega)^\\frac{2}{n}}k^\\frac{2}{n}\\qquad\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.1135","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:38:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bQspO2nviiYQ4lnV2rgcufoAoET8MEjmKNnwvHTuIVB35cNXe/wCgrhmYL2CyEWEp+2TGFAb8E7OHJgK/7euDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T07:25:24.129186Z"},"content_sha256":"c7069651be905b6b9a409c80e9a9b085d355f7779b621f263ce2376fb7ff7716","schema_version":"1.0","event_id":"sha256:c7069651be905b6b9a409c80e9a9b085d355f7779b621f263ce2376fb7ff7716"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JWPRRSL2ZL4K4SSL5SGTFX4MCX/bundle.json","state_url":"https://pith.science/pith/JWPRRSL2ZL4K4SSL5SGTFX4MCX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JWPRRSL2ZL4K4SSL5SGTFX4MCX/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-09T07:25:24Z","links":{"resolver":"https://pith.science/pith/JWPRRSL2ZL4K4SSL5SGTFX4MCX","bundle":"https://pith.science/pith/JWPRRSL2ZL4K4SSL5SGTFX4MCX/bundle.json","state":"https://pith.science/pith/JWPRRSL2ZL4K4SSL5SGTFX4MCX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JWPRRSL2ZL4K4SSL5SGTFX4MCX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:JWPRRSL2ZL4K4SSL5SGTFX4MCX","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":"9173f6718df50e0edcba27491a0080795345e912e06fa40db8ee57730b621589","cross_cats_sorted":["math.AP","math.SP"],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.DG","submitted_at":"2014-11-05T03:06:46Z","title_canon_sha256":"df35a48c918b134b27acc51d54ecfc2fd6986f16016d9cc66fa2b1924eac6cb7"},"schema_version":"1.0","source":{"id":"1411.1135","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1411.1135","created_at":"2026-05-18T02:38:08Z"},{"alias_kind":"arxiv_version","alias_value":"1411.1135v2","created_at":"2026-05-18T02:38:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.1135","created_at":"2026-05-18T02:38:08Z"},{"alias_kind":"pith_short_12","alias_value":"JWPRRSL2ZL4K","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_16","alias_value":"JWPRRSL2ZL4K4SSL","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_8","alias_value":"JWPRRSL2","created_at":"2026-05-18T12:28:35Z"}],"graph_snapshots":[{"event_id":"sha256:c7069651be905b6b9a409c80e9a9b085d355f7779b621f263ce2376fb7ff7716","target":"graph","created_at":"2026-05-18T02:38:08Z","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"},"paper":{"abstract_excerpt":"In this paper, we study lower bounds for higher eigenvalues of the Dirichlet eigenvalue problem of the Laplacian on a bounded domain $\\Omega$ in $\\mathbb{R}^n$. It is well known that the $k$-th Dirichlet eigenvalue $\\lambda_k$ obeys the Weyl asymptotic formula, that is,\n\\[\n\\lambda_k\\sim\\frac{4\\pi^2}{(\\omega_n\\mathrm{vol}\\Omega)^\\frac{2}{n}}k^\\frac{2}{n}\\qquad\\hbox{as}\\quad k\\rightarrow\\infty,\n\\]\nwhere $\\mathrm{vol}\\Omega$ is the volume of $\\Omega$. In view of the above formula, P\\'{o}lya conjectured that\n\\[\n\\lambda_k\\gs\\frac{4\\pi^2}{(\\omega_n\\mathrm{vol}\\Omega)^\\frac{2}{n}}k^\\frac{2}{n}\\qquad\\","authors_text":"Yue He","cross_cats":["math.AP","math.SP"],"headline":"","license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.DG","submitted_at":"2014-11-05T03:06:46Z","title":"Proof of the P\\'{o}lya conjecture"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.1135","kind":"arxiv","version":2},"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":"sha256:02ba3e8ddbbe3aa140d796727c757e88b17843a1549e0d2cd6184cfa2c257190","target":"record","created_at":"2026-05-18T02:38:08Z","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":"9173f6718df50e0edcba27491a0080795345e912e06fa40db8ee57730b621589","cross_cats_sorted":["math.AP","math.SP"],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.DG","submitted_at":"2014-11-05T03:06:46Z","title_canon_sha256":"df35a48c918b134b27acc51d54ecfc2fd6986f16016d9cc66fa2b1924eac6cb7"},"schema_version":"1.0","source":{"id":"1411.1135","kind":"arxiv","version":2}},"canonical_sha256":"4d9f18c97acaf8ae4a4bec8d32df8c15cf532ef404b6110990ca85ab0a9b59a6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4d9f18c97acaf8ae4a4bec8d32df8c15cf532ef404b6110990ca85ab0a9b59a6","first_computed_at":"2026-05-18T02:38:08.766456Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:38:08.766456Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DC5yNpgEaLV4i93oSD2fZhV0GWdhfrK0kRcaqIEfpY1CV0WXrP4i1fBxoR/KuSDYW+U7ORTWedvNutjMRPD/CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:38:08.767093Z","signed_message":"canonical_sha256_bytes"},"source_id":"1411.1135","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:02ba3e8ddbbe3aa140d796727c757e88b17843a1549e0d2cd6184cfa2c257190","sha256:c7069651be905b6b9a409c80e9a9b085d355f7779b621f263ce2376fb7ff7716"],"state_sha256":"cdbe35f94fe28f3228d8e452368d6c18fa5965a5f56749e9a54dbb76ee39dfc4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cmGW7v8k+QuINSzJbapFa8lozKnaZ+dSuFLL45DiiqthrUWBt5DO+/URsG1btnVYjm0m1JIN5NMxp/6BrSk4CA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T07:25:24.132729Z","bundle_sha256":"dc5f0e7d807c03f0d92219d16a1f2d777f5ff984da8facb42a040fa996f0b9d9"}}