{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:TH5G6JJHNJSEKN5MTSRZL3VVDX","short_pith_number":"pith:TH5G6JJH","canonical_record":{"source":{"id":"2606.08650","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-06-07T14:32:12Z","cross_cats_sorted":["math.CA","math.NT","math.SP"],"title_canon_sha256":"500f7a23ed550aa72b6aed2d40ccc7974cb37d54ee7681a87cb0cb57849dbf51","abstract_canon_sha256":"5d30e4a5b722f1101df95fc624f37c84aeaa3a7e16dbddf74ca594fa1ed6a369"},"schema_version":"1.0"},"canonical_sha256":"99fa6f25276a644537ac9ca395eeb51dcb7ffdbda19b2b7aecf686b28a1dc910","source":{"kind":"arxiv","id":"2606.08650","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.08650","created_at":"2026-06-09T01:05:42Z"},{"alias_kind":"arxiv_version","alias_value":"2606.08650v1","created_at":"2026-06-09T01:05:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.08650","created_at":"2026-06-09T01:05:42Z"},{"alias_kind":"pith_short_12","alias_value":"TH5G6JJHNJSE","created_at":"2026-06-09T01:05:42Z"},{"alias_kind":"pith_short_16","alias_value":"TH5G6JJHNJSEKN5M","created_at":"2026-06-09T01:05:42Z"},{"alias_kind":"pith_short_8","alias_value":"TH5G6JJH","created_at":"2026-06-09T01:05:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:TH5G6JJHNJSEKN5MTSRZL3VVDX","target":"record","payload":{"canonical_record":{"source":{"id":"2606.08650","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-06-07T14:32:12Z","cross_cats_sorted":["math.CA","math.NT","math.SP"],"title_canon_sha256":"500f7a23ed550aa72b6aed2d40ccc7974cb37d54ee7681a87cb0cb57849dbf51","abstract_canon_sha256":"5d30e4a5b722f1101df95fc624f37c84aeaa3a7e16dbddf74ca594fa1ed6a369"},"schema_version":"1.0"},"canonical_sha256":"99fa6f25276a644537ac9ca395eeb51dcb7ffdbda19b2b7aecf686b28a1dc910","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-09T01:05:42.711651Z","signature_b64":"shulA99sZwY9VjNj2FijmHMhxXkDt7+BIFmsm5d4XEpz5dMMXdMjeurfdcGZ2zWwgJMYHp8wRGr1otchkID1Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"99fa6f25276a644537ac9ca395eeb51dcb7ffdbda19b2b7aecf686b28a1dc910","last_reissued_at":"2026-06-09T01:05:42.711218Z","signature_status":"signed_v1","first_computed_at":"2026-06-09T01:05:42.711218Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.08650","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":"2026-06-09T01:05:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"22fyXV1JNE26GI/G3A8IN9BFu1YukfeKTFxPN7HZpggofyh8MxG6hETVbKMWscsdseBH+c3nnrkUZ621WDA4CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T23:48:16.807948Z"},"content_sha256":"6229e2cf73a77018bb6ef5b07a64898d6ab0cadb11fb4dc7696c33197a153e29","schema_version":"1.0","event_id":"sha256:6229e2cf73a77018bb6ef5b07a64898d6ab0cadb11fb4dc7696c33197a153e29"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:TH5G6JJHNJSEKN5MTSRZL3VVDX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Restriction estimates for toral eigenfunctions and lattice points in spherical regions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CA","math.NT","math.SP"],"primary_cat":"math.AP","authors_text":"Cheng Zhang, Zhifei Zhu","submitted_at":"2026-06-07T14:32:12Z","abstract_excerpt":"We establish new $L^2$ restriction estimates for toral eigenfunctions. These estimates are sharp in certain cases, and thus prove a conjecture of Huang-Zhang for smooth submanifolds of large codimension. In particular, they provide new progress toward a conjecture of Bourgain-Rudnick. The proof combines a slicing and packing method with the approximation of the discrete spherical multiplier by Magyar-Stein-Wainger and Magyar."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.08650","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/2606.08650/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":"2026-06-09T01:05:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9du/8puHLRaRCOYgMw1TO/2DfgJ+trWX9fdIGGFl4y8FhZQLlJwkHYjcVQKAemdsfXr5cdGTHXMIo9lR8wP1Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T23:48:16.808343Z"},"content_sha256":"703b833fe0b582da23f567e0b6f7f49caddecaf0caa05ea3a3ed05b44e69bd10","schema_version":"1.0","event_id":"sha256:703b833fe0b582da23f567e0b6f7f49caddecaf0caa05ea3a3ed05b44e69bd10"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TH5G6JJHNJSEKN5MTSRZL3VVDX/bundle.json","state_url":"https://pith.science/pith/TH5G6JJHNJSEKN5MTSRZL3VVDX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TH5G6JJHNJSEKN5MTSRZL3VVDX/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":"2026-06-21T23:48:16Z","links":{"resolver":"https://pith.science/pith/TH5G6JJHNJSEKN5MTSRZL3VVDX","bundle":"https://pith.science/pith/TH5G6JJHNJSEKN5MTSRZL3VVDX/bundle.json","state":"https://pith.science/pith/TH5G6JJHNJSEKN5MTSRZL3VVDX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TH5G6JJHNJSEKN5MTSRZL3VVDX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:TH5G6JJHNJSEKN5MTSRZL3VVDX","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":"5d30e4a5b722f1101df95fc624f37c84aeaa3a7e16dbddf74ca594fa1ed6a369","cross_cats_sorted":["math.CA","math.NT","math.SP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-06-07T14:32:12Z","title_canon_sha256":"500f7a23ed550aa72b6aed2d40ccc7974cb37d54ee7681a87cb0cb57849dbf51"},"schema_version":"1.0","source":{"id":"2606.08650","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.08650","created_at":"2026-06-09T01:05:42Z"},{"alias_kind":"arxiv_version","alias_value":"2606.08650v1","created_at":"2026-06-09T01:05:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.08650","created_at":"2026-06-09T01:05:42Z"},{"alias_kind":"pith_short_12","alias_value":"TH5G6JJHNJSE","created_at":"2026-06-09T01:05:42Z"},{"alias_kind":"pith_short_16","alias_value":"TH5G6JJHNJSEKN5M","created_at":"2026-06-09T01:05:42Z"},{"alias_kind":"pith_short_8","alias_value":"TH5G6JJH","created_at":"2026-06-09T01:05:42Z"}],"graph_snapshots":[{"event_id":"sha256:703b833fe0b582da23f567e0b6f7f49caddecaf0caa05ea3a3ed05b44e69bd10","target":"graph","created_at":"2026-06-09T01:05:42Z","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/2606.08650/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We establish new $L^2$ restriction estimates for toral eigenfunctions. These estimates are sharp in certain cases, and thus prove a conjecture of Huang-Zhang for smooth submanifolds of large codimension. In particular, they provide new progress toward a conjecture of Bourgain-Rudnick. The proof combines a slicing and packing method with the approximation of the discrete spherical multiplier by Magyar-Stein-Wainger and Magyar.","authors_text":"Cheng Zhang, Zhifei Zhu","cross_cats":["math.CA","math.NT","math.SP"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-06-07T14:32:12Z","title":"Restriction estimates for toral eigenfunctions and lattice points in spherical regions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.08650","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":"sha256:6229e2cf73a77018bb6ef5b07a64898d6ab0cadb11fb4dc7696c33197a153e29","target":"record","created_at":"2026-06-09T01:05:42Z","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":"5d30e4a5b722f1101df95fc624f37c84aeaa3a7e16dbddf74ca594fa1ed6a369","cross_cats_sorted":["math.CA","math.NT","math.SP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-06-07T14:32:12Z","title_canon_sha256":"500f7a23ed550aa72b6aed2d40ccc7974cb37d54ee7681a87cb0cb57849dbf51"},"schema_version":"1.0","source":{"id":"2606.08650","kind":"arxiv","version":1}},"canonical_sha256":"99fa6f25276a644537ac9ca395eeb51dcb7ffdbda19b2b7aecf686b28a1dc910","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"99fa6f25276a644537ac9ca395eeb51dcb7ffdbda19b2b7aecf686b28a1dc910","first_computed_at":"2026-06-09T01:05:42.711218Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-09T01:05:42.711218Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"shulA99sZwY9VjNj2FijmHMhxXkDt7+BIFmsm5d4XEpz5dMMXdMjeurfdcGZ2zWwgJMYHp8wRGr1otchkID1Cg==","signature_status":"signed_v1","signed_at":"2026-06-09T01:05:42.711651Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.08650","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6229e2cf73a77018bb6ef5b07a64898d6ab0cadb11fb4dc7696c33197a153e29","sha256:703b833fe0b582da23f567e0b6f7f49caddecaf0caa05ea3a3ed05b44e69bd10"],"state_sha256":"91644450afc4c1069b5a6ea472b8783b20c1611306e2adb8ed9d6c10f07d173c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0NCVbEp0g1mvRvtPpw+n47sDZDXAKl4YH+N8tLuagFSAdzBsX3iD53I8nCAQTaitSUkpWQXgwWj55t86GqbTAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T23:48:16.810413Z","bundle_sha256":"eda11d4ab32dcfbc6056f5149c0d2bc7ee47633245f9097e462b12ff6ad00884"}}