{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:ENQGRIVOBZEXX2ZOCKSR56IPPM","short_pith_number":"pith:ENQGRIVO","canonical_record":{"source":{"id":"1302.5487","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-02-22T05:35:20Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"632af9f3a273735db16efe2aecfb7f123bc860ce8a676341840a6a91942df8b0","abstract_canon_sha256":"8241400c5336b356e652a652d544e648a1262d24e081167f07db9aed4e2face7"},"schema_version":"1.0"},"canonical_sha256":"236068a2ae0e497beb2e12a51ef90f7b35a9bd7228ac7030e087a75dce0c0aba","source":{"kind":"arxiv","id":"1302.5487","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1302.5487","created_at":"2026-05-18T03:32:46Z"},{"alias_kind":"arxiv_version","alias_value":"1302.5487v1","created_at":"2026-05-18T03:32:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.5487","created_at":"2026-05-18T03:32:46Z"},{"alias_kind":"pith_short_12","alias_value":"ENQGRIVOBZEX","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"ENQGRIVOBZEXX2ZO","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"ENQGRIVO","created_at":"2026-05-18T12:27:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:ENQGRIVOBZEXX2ZOCKSR56IPPM","target":"record","payload":{"canonical_record":{"source":{"id":"1302.5487","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-02-22T05:35:20Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"632af9f3a273735db16efe2aecfb7f123bc860ce8a676341840a6a91942df8b0","abstract_canon_sha256":"8241400c5336b356e652a652d544e648a1262d24e081167f07db9aed4e2face7"},"schema_version":"1.0"},"canonical_sha256":"236068a2ae0e497beb2e12a51ef90f7b35a9bd7228ac7030e087a75dce0c0aba","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:32:46.675618Z","signature_b64":"3SmKg3XMUHxDleySbj4Mwpl5erBNX3MW258/OVOW9QGFFALvu/A7HHf+tBkppMe8DU3UmYaTOoiwc6Yk2fleDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"236068a2ae0e497beb2e12a51ef90f7b35a9bd7228ac7030e087a75dce0c0aba","last_reissued_at":"2026-05-18T03:32:46.674968Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:32:46.674968Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1302.5487","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":"2026-05-18T03:32:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BBnEkCdtPdMel3j5n1L7eOE31w4oFGadcnEIxM4by4PcnKtAyOX/784tf7faayzZHJv3NKStkVkFNE9aw02wDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T16:11:30.836248Z"},"content_sha256":"1e28751aae4e535be4885090a9ca1121e860ab51a48aa3607564358c9487064b","schema_version":"1.0","event_id":"sha256:1e28751aae4e535be4885090a9ca1121e860ab51a48aa3607564358c9487064b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:ENQGRIVOBZEXX2ZOCKSR56IPPM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Stable phase retrieval with low-redundancy frames","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.FA","authors_text":"Bernhard G. Bodmann, Nathaniel Hammen","submitted_at":"2013-02-22T05:35:20Z","abstract_excerpt":"We investigate the recovery of vectors from magnitudes of frame coefficients when the frames have a low redundancy, meaning a small number of frame vectors compared to the dimension of the Hilbert space. We first show that for vectors in d dimensions, 4d-4 suitably chosen frame vectors are sufficient to uniquely determine each signal, up to an overall unimodular constant, from the magnitudes of its frame coefficients. Then we discuss the effect of noise and show that 8d-4 frame vectors provide a stable recovery if part of the frame coefficients is bounded away from zero. In this regime, pertur"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.5487","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":"2026-05-18T03:32:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6XP8tYBr3ne9zRwIlYIVKBGGt9M5WoI/PPINJbc78yAYOOV/V74RC8YFhzL+Q64Y9w+WhCETZHnLj90qBVGYAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T16:11:30.836612Z"},"content_sha256":"980d1afb1881c07bef328950b83dd1c18fc2ea63685e6e772f61660c1a3fc3aa","schema_version":"1.0","event_id":"sha256:980d1afb1881c07bef328950b83dd1c18fc2ea63685e6e772f61660c1a3fc3aa"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ENQGRIVOBZEXX2ZOCKSR56IPPM/bundle.json","state_url":"https://pith.science/pith/ENQGRIVOBZEXX2ZOCKSR56IPPM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ENQGRIVOBZEXX2ZOCKSR56IPPM/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":"2026-06-02T16:11:30Z","links":{"resolver":"https://pith.science/pith/ENQGRIVOBZEXX2ZOCKSR56IPPM","bundle":"https://pith.science/pith/ENQGRIVOBZEXX2ZOCKSR56IPPM/bundle.json","state":"https://pith.science/pith/ENQGRIVOBZEXX2ZOCKSR56IPPM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ENQGRIVOBZEXX2ZOCKSR56IPPM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:ENQGRIVOBZEXX2ZOCKSR56IPPM","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":"8241400c5336b356e652a652d544e648a1262d24e081167f07db9aed4e2face7","cross_cats_sorted":["math.CV"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-02-22T05:35:20Z","title_canon_sha256":"632af9f3a273735db16efe2aecfb7f123bc860ce8a676341840a6a91942df8b0"},"schema_version":"1.0","source":{"id":"1302.5487","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1302.5487","created_at":"2026-05-18T03:32:46Z"},{"alias_kind":"arxiv_version","alias_value":"1302.5487v1","created_at":"2026-05-18T03:32:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.5487","created_at":"2026-05-18T03:32:46Z"},{"alias_kind":"pith_short_12","alias_value":"ENQGRIVOBZEX","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"ENQGRIVOBZEXX2ZO","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"ENQGRIVO","created_at":"2026-05-18T12:27:43Z"}],"graph_snapshots":[{"event_id":"sha256:980d1afb1881c07bef328950b83dd1c18fc2ea63685e6e772f61660c1a3fc3aa","target":"graph","created_at":"2026-05-18T03:32:46Z","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 investigate the recovery of vectors from magnitudes of frame coefficients when the frames have a low redundancy, meaning a small number of frame vectors compared to the dimension of the Hilbert space. We first show that for vectors in d dimensions, 4d-4 suitably chosen frame vectors are sufficient to uniquely determine each signal, up to an overall unimodular constant, from the magnitudes of its frame coefficients. Then we discuss the effect of noise and show that 8d-4 frame vectors provide a stable recovery if part of the frame coefficients is bounded away from zero. In this regime, pertur","authors_text":"Bernhard G. Bodmann, Nathaniel Hammen","cross_cats":["math.CV"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-02-22T05:35:20Z","title":"Stable phase retrieval with low-redundancy frames"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.5487","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":"sha256:1e28751aae4e535be4885090a9ca1121e860ab51a48aa3607564358c9487064b","target":"record","created_at":"2026-05-18T03:32:46Z","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":"8241400c5336b356e652a652d544e648a1262d24e081167f07db9aed4e2face7","cross_cats_sorted":["math.CV"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-02-22T05:35:20Z","title_canon_sha256":"632af9f3a273735db16efe2aecfb7f123bc860ce8a676341840a6a91942df8b0"},"schema_version":"1.0","source":{"id":"1302.5487","kind":"arxiv","version":1}},"canonical_sha256":"236068a2ae0e497beb2e12a51ef90f7b35a9bd7228ac7030e087a75dce0c0aba","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"236068a2ae0e497beb2e12a51ef90f7b35a9bd7228ac7030e087a75dce0c0aba","first_computed_at":"2026-05-18T03:32:46.674968Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:32:46.674968Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3SmKg3XMUHxDleySbj4Mwpl5erBNX3MW258/OVOW9QGFFALvu/A7HHf+tBkppMe8DU3UmYaTOoiwc6Yk2fleDw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:32:46.675618Z","signed_message":"canonical_sha256_bytes"},"source_id":"1302.5487","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1e28751aae4e535be4885090a9ca1121e860ab51a48aa3607564358c9487064b","sha256:980d1afb1881c07bef328950b83dd1c18fc2ea63685e6e772f61660c1a3fc3aa"],"state_sha256":"6ea66e85e5b33087a6548d120db8befc04fb28c74c2a3bb91f4218846a727d81"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8Id0aziZCSjcu32QErB5XJRBa4HR1hu9idsqnfm7XmfNNspwFSMsexfp3b34Ee5G4NhtEQPE81FBEBi8PykpAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T16:11:30.838726Z","bundle_sha256":"327939dbd6679f65b3cdfa6f00d8eada9696de99ca288347a206666f047a176e"}}