{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:7BSWH76LK5FNVBRHPNPZK6NK2U","short_pith_number":"pith:7BSWH76L","canonical_record":{"source":{"id":"1204.3306","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-04-15T18:56:22Z","cross_cats_sorted":["math.NA"],"title_canon_sha256":"79710b82821a8d6916f1798db70ee8b0cc9e6ce500e90656abda3a4c99dbeb89","abstract_canon_sha256":"2238a4ce59f01228d1f3c39a4cbe06768c29e14d727f275940b1a369fecee4ad"},"schema_version":"1.0"},"canonical_sha256":"f86563ffcb574ada86277b5f9579aad5047daa14a023b1b51716106c7fc13344","source":{"kind":"arxiv","id":"1204.3306","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.3306","created_at":"2026-05-18T03:57:51Z"},{"alias_kind":"arxiv_version","alias_value":"1204.3306v1","created_at":"2026-05-18T03:57:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.3306","created_at":"2026-05-18T03:57:51Z"},{"alias_kind":"pith_short_12","alias_value":"7BSWH76LK5FN","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_16","alias_value":"7BSWH76LK5FNVBRH","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_8","alias_value":"7BSWH76L","created_at":"2026-05-18T12:26:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:7BSWH76LK5FNVBRHPNPZK6NK2U","target":"record","payload":{"canonical_record":{"source":{"id":"1204.3306","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-04-15T18:56:22Z","cross_cats_sorted":["math.NA"],"title_canon_sha256":"79710b82821a8d6916f1798db70ee8b0cc9e6ce500e90656abda3a4c99dbeb89","abstract_canon_sha256":"2238a4ce59f01228d1f3c39a4cbe06768c29e14d727f275940b1a369fecee4ad"},"schema_version":"1.0"},"canonical_sha256":"f86563ffcb574ada86277b5f9579aad5047daa14a023b1b51716106c7fc13344","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:57:51.883141Z","signature_b64":"aYqeqBx/Civg3l6CJp1jASdvsdMPGjyrV2ygzvecY4rwgvELJU1NeFW/J0YmV2jPRBYXXb7qE0fi1QFNc5X4BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f86563ffcb574ada86277b5f9579aad5047daa14a023b1b51716106c7fc13344","last_reissued_at":"2026-05-18T03:57:51.882519Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:57:51.882519Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1204.3306","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":"2026-05-18T03:57:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"E22vkCldQgf5igliDNPRwNXJjKRsPLTfizlg/gLkS4ggmRQK4WrxAUCVpPD6rkYSiYJzB52k29LwDZpPI3QQBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T11:53:43.942983Z"},"content_sha256":"ee459c3195e42fbf4f516aa4a275dbe5a92c47e98ab89d51160a0d2ef7f374a8","schema_version":"1.0","event_id":"sha256:ee459c3195e42fbf4f516aa4a275dbe5a92c47e98ab89d51160a0d2ef7f374a8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:7BSWH76LK5FNVBRHPNPZK6NK2U","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Necessary and sufficient conditions to perform Spectral Tetris","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NA"],"primary_cat":"math.FA","authors_text":"Andreas Heinecke, Keri Kornelson, Peter Casazza, Yang Wang, Zhengfang Zhou","submitted_at":"2012-04-15T18:56:22Z","abstract_excerpt":"Spectral Tetris has proved to be a powerful tool for constructing sparse equal norm Hilbert space frames. We introduce a new form of Spectral Tetris which works for non-equal norm frames. It is known that this method cannot construct all frames --- even in the new case introduced here. Until now, it has been a mystery as to why Spectral Tetris sometimes works and sometimes fails. We will give a complete answer to this mystery by giving necessary and sufficient conditions for Spectral Tetris to construct frames in all cases including equal norm frames, prescribed norm frames, frames with consta"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.3306","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":"2026-05-18T03:57:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IcWNSs7wCq4acqJMmBRjgwfH8KmWrMPR8jLf2XnyLjxUjZ/qgrQTgPa+oaP7AfBIJIQf3ieZGvQjVLZP6mmSAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T11:53:43.943342Z"},"content_sha256":"9bee10a441ba0ea3421230aea2af4a1f46a8346d1668c79cbf22257cdb9eac6e","schema_version":"1.0","event_id":"sha256:9bee10a441ba0ea3421230aea2af4a1f46a8346d1668c79cbf22257cdb9eac6e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7BSWH76LK5FNVBRHPNPZK6NK2U/bundle.json","state_url":"https://pith.science/pith/7BSWH76LK5FNVBRHPNPZK6NK2U/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7BSWH76LK5FNVBRHPNPZK6NK2U/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":"2026-05-28T11:53:43Z","links":{"resolver":"https://pith.science/pith/7BSWH76LK5FNVBRHPNPZK6NK2U","bundle":"https://pith.science/pith/7BSWH76LK5FNVBRHPNPZK6NK2U/bundle.json","state":"https://pith.science/pith/7BSWH76LK5FNVBRHPNPZK6NK2U/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7BSWH76LK5FNVBRHPNPZK6NK2U/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:7BSWH76LK5FNVBRHPNPZK6NK2U","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":"2238a4ce59f01228d1f3c39a4cbe06768c29e14d727f275940b1a369fecee4ad","cross_cats_sorted":["math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-04-15T18:56:22Z","title_canon_sha256":"79710b82821a8d6916f1798db70ee8b0cc9e6ce500e90656abda3a4c99dbeb89"},"schema_version":"1.0","source":{"id":"1204.3306","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.3306","created_at":"2026-05-18T03:57:51Z"},{"alias_kind":"arxiv_version","alias_value":"1204.3306v1","created_at":"2026-05-18T03:57:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.3306","created_at":"2026-05-18T03:57:51Z"},{"alias_kind":"pith_short_12","alias_value":"7BSWH76LK5FN","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_16","alias_value":"7BSWH76LK5FNVBRH","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_8","alias_value":"7BSWH76L","created_at":"2026-05-18T12:26:56Z"}],"graph_snapshots":[{"event_id":"sha256:9bee10a441ba0ea3421230aea2af4a1f46a8346d1668c79cbf22257cdb9eac6e","target":"graph","created_at":"2026-05-18T03:57:51Z","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":"Spectral Tetris has proved to be a powerful tool for constructing sparse equal norm Hilbert space frames. We introduce a new form of Spectral Tetris which works for non-equal norm frames. It is known that this method cannot construct all frames --- even in the new case introduced here. Until now, it has been a mystery as to why Spectral Tetris sometimes works and sometimes fails. We will give a complete answer to this mystery by giving necessary and sufficient conditions for Spectral Tetris to construct frames in all cases including equal norm frames, prescribed norm frames, frames with consta","authors_text":"Andreas Heinecke, Keri Kornelson, Peter Casazza, Yang Wang, Zhengfang Zhou","cross_cats":["math.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-04-15T18:56:22Z","title":"Necessary and sufficient conditions to perform Spectral Tetris"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.3306","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:ee459c3195e42fbf4f516aa4a275dbe5a92c47e98ab89d51160a0d2ef7f374a8","target":"record","created_at":"2026-05-18T03:57:51Z","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":"2238a4ce59f01228d1f3c39a4cbe06768c29e14d727f275940b1a369fecee4ad","cross_cats_sorted":["math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-04-15T18:56:22Z","title_canon_sha256":"79710b82821a8d6916f1798db70ee8b0cc9e6ce500e90656abda3a4c99dbeb89"},"schema_version":"1.0","source":{"id":"1204.3306","kind":"arxiv","version":1}},"canonical_sha256":"f86563ffcb574ada86277b5f9579aad5047daa14a023b1b51716106c7fc13344","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f86563ffcb574ada86277b5f9579aad5047daa14a023b1b51716106c7fc13344","first_computed_at":"2026-05-18T03:57:51.882519Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:57:51.882519Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aYqeqBx/Civg3l6CJp1jASdvsdMPGjyrV2ygzvecY4rwgvELJU1NeFW/J0YmV2jPRBYXXb7qE0fi1QFNc5X4BA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:57:51.883141Z","signed_message":"canonical_sha256_bytes"},"source_id":"1204.3306","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ee459c3195e42fbf4f516aa4a275dbe5a92c47e98ab89d51160a0d2ef7f374a8","sha256:9bee10a441ba0ea3421230aea2af4a1f46a8346d1668c79cbf22257cdb9eac6e"],"state_sha256":"894995fca17cc5dfb132b90bf6f73a1d8908d1689c156e3959a885fb5819b03f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xF0GtJZ9sVp7Zd3pRVS6fGc4lRbadZv0bRX4HqVMZXXqZ5sdYF935pOX1qHLWxkO7FTo0/ZRRFKawwGQ+SwhAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T11:53:43.945354Z","bundle_sha256":"4aa1eec2d649842fa876a8358a471910519c0350d94ea93a8e90d1f470d2aeac"}}