{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:BTUKIQ45I5ZAEZDECZEKQKRW6I","short_pith_number":"pith:BTUKIQ45","canonical_record":{"source":{"id":"1501.05257","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-01-21T18:52:10Z","cross_cats_sorted":[],"title_canon_sha256":"2a7ef96c136071f139901fdb7fb1aa078250de5712cb25adeff30d0c5541c18f","abstract_canon_sha256":"46ee619a61df1bab726d8aafa01cefa95b5e6f25e6fbb7cdbfbb3c986f2856b8"},"schema_version":"1.0"},"canonical_sha256":"0ce8a4439d47720264641648a82a36f216470228c943702ba6556c7a3b35adf9","source":{"kind":"arxiv","id":"1501.05257","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.05257","created_at":"2026-05-18T01:41:09Z"},{"alias_kind":"arxiv_version","alias_value":"1501.05257v1","created_at":"2026-05-18T01:41:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.05257","created_at":"2026-05-18T01:41:09Z"},{"alias_kind":"pith_short_12","alias_value":"BTUKIQ45I5ZA","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_16","alias_value":"BTUKIQ45I5ZAEZDE","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_8","alias_value":"BTUKIQ45","created_at":"2026-05-18T12:29:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:BTUKIQ45I5ZAEZDECZEKQKRW6I","target":"record","payload":{"canonical_record":{"source":{"id":"1501.05257","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-01-21T18:52:10Z","cross_cats_sorted":[],"title_canon_sha256":"2a7ef96c136071f139901fdb7fb1aa078250de5712cb25adeff30d0c5541c18f","abstract_canon_sha256":"46ee619a61df1bab726d8aafa01cefa95b5e6f25e6fbb7cdbfbb3c986f2856b8"},"schema_version":"1.0"},"canonical_sha256":"0ce8a4439d47720264641648a82a36f216470228c943702ba6556c7a3b35adf9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:41:09.578351Z","signature_b64":"26sGiUtUapUi+rWEkGgJtfobyv+L7BCb8G3sCEn01SHbT2u86RFiz9ULCBvqtkT6e3Je/K/HwW7VAwfwo3NLDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0ce8a4439d47720264641648a82a36f216470228c943702ba6556c7a3b35adf9","last_reissued_at":"2026-05-18T01:41:09.577620Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:41:09.577620Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1501.05257","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":"2026-05-18T01:41:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Bppi89juvv3zKSvM/YjGyCaUhuLVMittmVUc+Wka1qsS+PnvmRAnIK0SpBjJeuE5a4/uRWNgko/XRa9NFe43DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T06:39:54.655597Z"},"content_sha256":"87193744e55c24e179e333c4e6f7b57e72a4f1450d826423161c31e20ca0dc7f","schema_version":"1.0","event_id":"sha256:87193744e55c24e179e333c4e6f7b57e72a4f1450d826423161c31e20ca0dc7f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:BTUKIQ45I5ZAEZDECZEKQKRW6I","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Combining Riesz bases in $R^d$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Gady Kozma, Shahaf Nitzan","submitted_at":"2015-01-21T18:52:10Z","abstract_excerpt":"We prove that every finite union of rectangles in $R^d$ admits a Riesz basis of exponentials."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.05257","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":""},"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":"2026-05-18T01:41:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jhJwfRP4PA1QCQf8CUhwf4tuUje5ZGl/eDsnlJAxdWyqPXQe++edEKwmuBehTQnNFMIATO9geApbe6axWixlAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T06:39:54.656241Z"},"content_sha256":"df277a9e8400797392855cdc825963fc2d990d0168c1b081f7bd7da80d7a01b5","schema_version":"1.0","event_id":"sha256:df277a9e8400797392855cdc825963fc2d990d0168c1b081f7bd7da80d7a01b5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BTUKIQ45I5ZAEZDECZEKQKRW6I/bundle.json","state_url":"https://pith.science/pith/BTUKIQ45I5ZAEZDECZEKQKRW6I/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BTUKIQ45I5ZAEZDECZEKQKRW6I/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":"2026-06-05T06:39:54Z","links":{"resolver":"https://pith.science/pith/BTUKIQ45I5ZAEZDECZEKQKRW6I","bundle":"https://pith.science/pith/BTUKIQ45I5ZAEZDECZEKQKRW6I/bundle.json","state":"https://pith.science/pith/BTUKIQ45I5ZAEZDECZEKQKRW6I/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BTUKIQ45I5ZAEZDECZEKQKRW6I/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:BTUKIQ45I5ZAEZDECZEKQKRW6I","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":"46ee619a61df1bab726d8aafa01cefa95b5e6f25e6fbb7cdbfbb3c986f2856b8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-01-21T18:52:10Z","title_canon_sha256":"2a7ef96c136071f139901fdb7fb1aa078250de5712cb25adeff30d0c5541c18f"},"schema_version":"1.0","source":{"id":"1501.05257","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.05257","created_at":"2026-05-18T01:41:09Z"},{"alias_kind":"arxiv_version","alias_value":"1501.05257v1","created_at":"2026-05-18T01:41:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.05257","created_at":"2026-05-18T01:41:09Z"},{"alias_kind":"pith_short_12","alias_value":"BTUKIQ45I5ZA","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_16","alias_value":"BTUKIQ45I5ZAEZDE","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_8","alias_value":"BTUKIQ45","created_at":"2026-05-18T12:29:14Z"}],"graph_snapshots":[{"event_id":"sha256:df277a9e8400797392855cdc825963fc2d990d0168c1b081f7bd7da80d7a01b5","target":"graph","created_at":"2026-05-18T01:41:09Z","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":"We prove that every finite union of rectangles in $R^d$ admits a Riesz basis of exponentials.","authors_text":"Gady Kozma, Shahaf Nitzan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-01-21T18:52:10Z","title":"Combining Riesz bases in $R^d$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.05257","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":"sha256:87193744e55c24e179e333c4e6f7b57e72a4f1450d826423161c31e20ca0dc7f","target":"record","created_at":"2026-05-18T01:41:09Z","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":"46ee619a61df1bab726d8aafa01cefa95b5e6f25e6fbb7cdbfbb3c986f2856b8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-01-21T18:52:10Z","title_canon_sha256":"2a7ef96c136071f139901fdb7fb1aa078250de5712cb25adeff30d0c5541c18f"},"schema_version":"1.0","source":{"id":"1501.05257","kind":"arxiv","version":1}},"canonical_sha256":"0ce8a4439d47720264641648a82a36f216470228c943702ba6556c7a3b35adf9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0ce8a4439d47720264641648a82a36f216470228c943702ba6556c7a3b35adf9","first_computed_at":"2026-05-18T01:41:09.577620Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:41:09.577620Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"26sGiUtUapUi+rWEkGgJtfobyv+L7BCb8G3sCEn01SHbT2u86RFiz9ULCBvqtkT6e3Je/K/HwW7VAwfwo3NLDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:41:09.578351Z","signed_message":"canonical_sha256_bytes"},"source_id":"1501.05257","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:87193744e55c24e179e333c4e6f7b57e72a4f1450d826423161c31e20ca0dc7f","sha256:df277a9e8400797392855cdc825963fc2d990d0168c1b081f7bd7da80d7a01b5"],"state_sha256":"d7fe0dc217d8788151d4c9702d392ac40baecaad4b36af017c2bed25fc5bd81a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GX2y2Ng5mimSQi56Hk3MD5cRJ59TQJM8pIs6Y9sJAyLND4ITvwPkgEAiybNyPdOMwZ46Wi0GBlbdQTOnFdHZCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T06:39:54.659837Z","bundle_sha256":"dfc1df614d90ee5a7daa86106b475861430b18742b994cf96abbdbe3e648e67b"}}