{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:XXLNRZBFDLDJ6LA7FSJ3QP7UNB","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":"77fb4ebb9c62743b3549bb391d8ae825a7225f029cbbcc31464745b54d24b5fe","cross_cats_sorted":["math.AP","math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2026-07-28T16:43:50Z","title_canon_sha256":"14b4086b59b70e5b13806b55a6352afc8b8c0a7a3f26e7aa90971617610c8544"},"schema_version":"1.0","source":{"id":"2607.25958","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.25958","created_at":"2026-07-29T01:26:13Z"},{"alias_kind":"arxiv_version","alias_value":"2607.25958v1","created_at":"2026-07-29T01:26:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.25958","created_at":"2026-07-29T01:26:13Z"},{"alias_kind":"pith_short_12","alias_value":"XXLNRZBFDLDJ","created_at":"2026-07-29T01:26:13Z"},{"alias_kind":"pith_short_16","alias_value":"XXLNRZBFDLDJ6LA7","created_at":"2026-07-29T01:26:13Z"},{"alias_kind":"pith_short_8","alias_value":"XXLNRZBF","created_at":"2026-07-29T01:26:13Z"}],"graph_snapshots":[{"event_id":"sha256:164d696f45c973519493a2c40e93d545e0413516bf4ec67f2eb151a5e50c4ef4","target":"graph","created_at":"2026-07-29T01:26:13Z","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/2607.25958/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove P\\'olya's conjectured lower bound for the Neumann counting function of every Euclidean ball. If $d\\ge2$, $R>0$, and $E\\ge0$, then \\begin{equation*} N_{B_R^d}^{<}(E) \\ge \\frac{\\omega_d}{(2\\pi)^d}|B_R^d|E^{d/2} = \\frac{(R\\sqrt E)^d}{2^d\\Gamma(\\frac d2+1)^2}. \\end{equation*} The radial boundary condition in dimensions $d\\ge3$ is a Dini condition, not a derivative-zero condition. A strict Robin comparison first reduces it to a Bessel phase estimate. Variational bounds handle low frequencies, while estimates based on finitely many radial levels and on beta moments cover the intermediate ra","authors_text":"Yutian Li","cross_cats":["math.AP","math.CA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2026-07-28T16:43:50Z","title":"P\\'olya's Conjecture for the Neumann Laplacian on Euclidean Balls"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.25958","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":"sha256:84d8a4207860a806a48f5a823cc1910eb70758e9f10d1022ec728c973d0cb468","target":"record","created_at":"2026-07-29T01:26:13Z","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":"77fb4ebb9c62743b3549bb391d8ae825a7225f029cbbcc31464745b54d24b5fe","cross_cats_sorted":["math.AP","math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2026-07-28T16:43:50Z","title_canon_sha256":"14b4086b59b70e5b13806b55a6352afc8b8c0a7a3f26e7aa90971617610c8544"},"schema_version":"1.0","source":{"id":"2607.25958","kind":"arxiv","version":1}},"canonical_sha256":"bdd6d8e4251ac69f2c1f2c93b83ff46844361fe437d9dc59beb98c9e840a9ce2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bdd6d8e4251ac69f2c1f2c93b83ff46844361fe437d9dc59beb98c9e840a9ce2","first_computed_at":"2026-07-29T01:26:13.073449Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-29T01:26:13.073449Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rFwsjhq4FP15MVh24pmdgUkrBoS3PNtIVz4jeOTxQ9B6bJ7uZhs6e6nyxvwqxJLo0RlF8T9S/dllO3RKAsQMCg==","signature_status":"signed_v1","signed_at":"2026-07-29T01:26:13.074234Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.25958","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:84d8a4207860a806a48f5a823cc1910eb70758e9f10d1022ec728c973d0cb468","sha256:164d696f45c973519493a2c40e93d545e0413516bf4ec67f2eb151a5e50c4ef4"],"state_sha256":"cfb2c0837d7ffc7b6ddfac24be920c6845586c0f75dc74e70bf4958122f6e806"}