{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:CS5R5HZYSKVQZH3WL75DIQ3A7I","short_pith_number":"pith:CS5R5HZY","canonical_record":{"source":{"id":"2506.02299","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-06-02T22:37:13Z","cross_cats_sorted":[],"title_canon_sha256":"70147730230b768998c7893b371c9e952f72ee1ce49373a0fbbb7b70f6ecb08d","abstract_canon_sha256":"ede545d4d3f34eb527fdf80669065b107e8c7d75528a38332af5bdbf932f2a72"},"schema_version":"1.0"},"canonical_sha256":"14bb1e9f3892ab0c9f765ffa344360fa29a85c297c158a72251be481d6875d44","source":{"kind":"arxiv","id":"2506.02299","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.02299","created_at":"2026-07-05T11:14:35Z"},{"alias_kind":"arxiv_version","alias_value":"2506.02299v1","created_at":"2026-07-05T11:14:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.02299","created_at":"2026-07-05T11:14:35Z"},{"alias_kind":"pith_short_12","alias_value":"CS5R5HZYSKVQ","created_at":"2026-07-05T11:14:35Z"},{"alias_kind":"pith_short_16","alias_value":"CS5R5HZYSKVQZH3W","created_at":"2026-07-05T11:14:35Z"},{"alias_kind":"pith_short_8","alias_value":"CS5R5HZY","created_at":"2026-07-05T11:14:35Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:CS5R5HZYSKVQZH3WL75DIQ3A7I","target":"record","payload":{"canonical_record":{"source":{"id":"2506.02299","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-06-02T22:37:13Z","cross_cats_sorted":[],"title_canon_sha256":"70147730230b768998c7893b371c9e952f72ee1ce49373a0fbbb7b70f6ecb08d","abstract_canon_sha256":"ede545d4d3f34eb527fdf80669065b107e8c7d75528a38332af5bdbf932f2a72"},"schema_version":"1.0"},"canonical_sha256":"14bb1e9f3892ab0c9f765ffa344360fa29a85c297c158a72251be481d6875d44","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:14:35.971596Z","signature_b64":"OONvQ7F/8VTrQyx78qCONc5a3SiYetmEyzp83cobuBWt0UUZqmD23mitL7ZHUJnAFBzaXRyfJXRyuCEQj+G0BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"14bb1e9f3892ab0c9f765ffa344360fa29a85c297c158a72251be481d6875d44","last_reissued_at":"2026-07-05T11:14:35.971101Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:14:35.971101Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2506.02299","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:14:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KdfQlM9d7nmgSSSpoZRZJ0IqYdPg6gt1ESg9mcGnJBqm+4uAgqn1i4ThBA/mFyzY3ptBkkr+y4paLrF6Tdw9BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T06:24:22.352777Z"},"content_sha256":"14d29fd286ee613ded2139a458a7e84e2a5e13876f850537c57eccce58ccb7fd","schema_version":"1.0","event_id":"sha256:14d29fd286ee613ded2139a458a7e84e2a5e13876f850537c57eccce58ccb7fd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:CS5R5HZYSKVQZH3WL75DIQ3A7I","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Weyl formula improvement for product of Zoll manifolds","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Yanfei Wang","submitted_at":"2025-06-02T22:37:13Z","abstract_excerpt":"Iosevich and Wyman have proved in ~\\cite{IoWy} that the remainder term in classical Weyl law can be improved from $O(\\lambda^{d-1})$ to $o(\\lambda^{d-1})$ in the case of product manifold by using a famous result of Duistermaat and Guillemin. They also showed that we could have polynomial improvement in the special case of Cartesian product of round spheres by reducing the problem to the study of the distribution of weighted integer lattice points. In this paper, we show that we can extend this result to the case of Cartesian product of Zoll manifolds by investigating the eigenvalue clusters of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.02299","kind":"arxiv","version":1},"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/2506.02299/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:14:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GvASUMZjxGIKTuRVKfqHIYgw/a0XL2YHHvoeMA4o2CQHlx8MgmZs2q90OmQUWpv6GL55jqjQrUydFNc0DqHMCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T06:24:22.353323Z"},"content_sha256":"267b16d508c0f8a7619f12c726ec8eaa6e8bb737ae6a1ccf2cfa9a0dd5a7651c","schema_version":"1.0","event_id":"sha256:267b16d508c0f8a7619f12c726ec8eaa6e8bb737ae6a1ccf2cfa9a0dd5a7651c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CS5R5HZYSKVQZH3WL75DIQ3A7I/bundle.json","state_url":"https://pith.science/pith/CS5R5HZYSKVQZH3WL75DIQ3A7I/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CS5R5HZYSKVQZH3WL75DIQ3A7I/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":"2026-08-08T06:24:22Z","links":{"resolver":"https://pith.science/pith/CS5R5HZYSKVQZH3WL75DIQ3A7I","bundle":"https://pith.science/pith/CS5R5HZYSKVQZH3WL75DIQ3A7I/bundle.json","state":"https://pith.science/pith/CS5R5HZYSKVQZH3WL75DIQ3A7I/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CS5R5HZYSKVQZH3WL75DIQ3A7I/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:CS5R5HZYSKVQZH3WL75DIQ3A7I","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":"ede545d4d3f34eb527fdf80669065b107e8c7d75528a38332af5bdbf932f2a72","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-06-02T22:37:13Z","title_canon_sha256":"70147730230b768998c7893b371c9e952f72ee1ce49373a0fbbb7b70f6ecb08d"},"schema_version":"1.0","source":{"id":"2506.02299","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.02299","created_at":"2026-07-05T11:14:35Z"},{"alias_kind":"arxiv_version","alias_value":"2506.02299v1","created_at":"2026-07-05T11:14:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.02299","created_at":"2026-07-05T11:14:35Z"},{"alias_kind":"pith_short_12","alias_value":"CS5R5HZYSKVQ","created_at":"2026-07-05T11:14:35Z"},{"alias_kind":"pith_short_16","alias_value":"CS5R5HZYSKVQZH3W","created_at":"2026-07-05T11:14:35Z"},{"alias_kind":"pith_short_8","alias_value":"CS5R5HZY","created_at":"2026-07-05T11:14:35Z"}],"graph_snapshots":[{"event_id":"sha256:267b16d508c0f8a7619f12c726ec8eaa6e8bb737ae6a1ccf2cfa9a0dd5a7651c","target":"graph","created_at":"2026-07-05T11:14:35Z","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/2506.02299/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Iosevich and Wyman have proved in ~\\cite{IoWy} that the remainder term in classical Weyl law can be improved from $O(\\lambda^{d-1})$ to $o(\\lambda^{d-1})$ in the case of product manifold by using a famous result of Duistermaat and Guillemin. They also showed that we could have polynomial improvement in the special case of Cartesian product of round spheres by reducing the problem to the study of the distribution of weighted integer lattice points. In this paper, we show that we can extend this result to the case of Cartesian product of Zoll manifolds by investigating the eigenvalue clusters of","authors_text":"Yanfei Wang","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-06-02T22:37:13Z","title":"Weyl formula improvement for product of Zoll manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.02299","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":"sha256:14d29fd286ee613ded2139a458a7e84e2a5e13876f850537c57eccce58ccb7fd","target":"record","created_at":"2026-07-05T11:14:35Z","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":"ede545d4d3f34eb527fdf80669065b107e8c7d75528a38332af5bdbf932f2a72","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-06-02T22:37:13Z","title_canon_sha256":"70147730230b768998c7893b371c9e952f72ee1ce49373a0fbbb7b70f6ecb08d"},"schema_version":"1.0","source":{"id":"2506.02299","kind":"arxiv","version":1}},"canonical_sha256":"14bb1e9f3892ab0c9f765ffa344360fa29a85c297c158a72251be481d6875d44","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"14bb1e9f3892ab0c9f765ffa344360fa29a85c297c158a72251be481d6875d44","first_computed_at":"2026-07-05T11:14:35.971101Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:14:35.971101Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OONvQ7F/8VTrQyx78qCONc5a3SiYetmEyzp83cobuBWt0UUZqmD23mitL7ZHUJnAFBzaXRyfJXRyuCEQj+G0BA==","signature_status":"signed_v1","signed_at":"2026-07-05T11:14:35.971596Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.02299","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:14d29fd286ee613ded2139a458a7e84e2a5e13876f850537c57eccce58ccb7fd","sha256:267b16d508c0f8a7619f12c726ec8eaa6e8bb737ae6a1ccf2cfa9a0dd5a7651c"],"state_sha256":"142a03402926fa325c7167c85cfadfea62cc9433bf132af428127b80e1f22e7a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iy0oTjFE0XCspqBYExylQD+UJxVxTfgYatcP2qS4FWf8uw2uy9f0siD9HxOlqxf9YEb5JEh/ayd5756ln0PnCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T06:24:22.358403Z","bundle_sha256":"ceadeb2cb0c93a79607eefaabded3cb45e999427f9e3358d6cc7686ae4c8a81d"}}