{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:EOOFI2ZZUD6J4MIX3HT4HGHVIK","short_pith_number":"pith:EOOFI2ZZ","canonical_record":{"source":{"id":"2607.06693","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.FA","submitted_at":"2026-07-07T18:10:15Z","cross_cats_sorted":["math.CA","math.PR"],"title_canon_sha256":"facaadb8135b0516f76efb40ce8789c81484056cc1e2b867b00f0418c9ae139e","abstract_canon_sha256":"84a098fe202487bb28814fb243451857fad4e17d153c9fcc38c22ef739b5f5ec"},"schema_version":"1.0"},"canonical_sha256":"239c546b39a0fc9e3117d9e7c398f542977c4a6d9895fe1eda779d15eefc451e","source":{"kind":"arxiv","id":"2607.06693","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.06693","created_at":"2026-07-09T00:19:23Z"},{"alias_kind":"arxiv_version","alias_value":"2607.06693v1","created_at":"2026-07-09T00:19:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.06693","created_at":"2026-07-09T00:19:23Z"},{"alias_kind":"pith_short_12","alias_value":"EOOFI2ZZUD6J","created_at":"2026-07-09T00:19:23Z"},{"alias_kind":"pith_short_16","alias_value":"EOOFI2ZZUD6J4MIX","created_at":"2026-07-09T00:19:23Z"},{"alias_kind":"pith_short_8","alias_value":"EOOFI2ZZ","created_at":"2026-07-09T00:19:23Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:EOOFI2ZZUD6J4MIX3HT4HGHVIK","target":"record","payload":{"canonical_record":{"source":{"id":"2607.06693","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.FA","submitted_at":"2026-07-07T18:10:15Z","cross_cats_sorted":["math.CA","math.PR"],"title_canon_sha256":"facaadb8135b0516f76efb40ce8789c81484056cc1e2b867b00f0418c9ae139e","abstract_canon_sha256":"84a098fe202487bb28814fb243451857fad4e17d153c9fcc38c22ef739b5f5ec"},"schema_version":"1.0"},"canonical_sha256":"239c546b39a0fc9e3117d9e7c398f542977c4a6d9895fe1eda779d15eefc451e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-09T00:19:23.688029Z","signature_b64":"NEFA7zysL4cx288ffCMify94tat5klvcXtWbrtWETD3/bYHhIXkAEbSuOJ+EtgkvYoFyyVPW2ieSb9fWOQWUAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"239c546b39a0fc9e3117d9e7c398f542977c4a6d9895fe1eda779d15eefc451e","last_reissued_at":"2026-07-09T00:19:23.687564Z","signature_status":"signed_v1","first_computed_at":"2026-07-09T00:19:23.687564Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.06693","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":"2026-07-09T00:19:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JNl9rmvJLTrcHz5wO4TpybVPOC80z0aMK3P2Xiog4fH2VElGWfmyxCOpjCzKV8GN97GDIvEs3PCOeHo5YymUBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T02:19:00.315598Z"},"content_sha256":"06eb2216d9dd13a12a5a3e70e28fb84b3d21fb929739b2b6b795082de06eed22","schema_version":"1.0","event_id":"sha256:06eb2216d9dd13a12a5a3e70e28fb84b3d21fb929739b2b6b795082de06eed22"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:EOOFI2ZZUD6J4MIX3HT4HGHVIK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Stable Phase Retrieval for Spans of Independent Random Variables","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CA","math.PR"],"primary_cat":"math.FA","authors_text":"Jaume de Dios Pont, Jo\\~ao P. G. Ramos, Mitchell A. Taylor, Pedro Abdalla","submitted_at":"2026-07-07T18:10:15Z","abstract_excerpt":"We prove that, after $L^2$ normalization, stable phase retrieval holds over the $L^2$-spans of independent real-valued centered random variables if and only if all but possibly one coordinate satisfies a uniform two-sided $L^1$ bound. This provides a complete characterization of stable phase retrieval for such subspaces, building upon the pioneering work of Calderbank--Daubechies--Freeman--Freeman and confirming the conjectured characterization communicated to us by those authors.\n  We provide two different proofs of this fact, both based on a decomposition of the $\\ell^2$-coefficients of each"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.06693","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/2607.06693/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":"2026-07-09T00:19:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uDqjbOfLQqmag9CbF2CidFQsT7E41IeAg6GzDa5mWHl9AqBFzCq5ABCFjugEe91Nkoxlk5CP0lkoyv+HRNsTBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T02:19:00.316285Z"},"content_sha256":"79fd6297dda8af192ff9a4dff0e55b40e2d024aa451ce8ebf171bbeb50dafa5d","schema_version":"1.0","event_id":"sha256:79fd6297dda8af192ff9a4dff0e55b40e2d024aa451ce8ebf171bbeb50dafa5d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EOOFI2ZZUD6J4MIX3HT4HGHVIK/bundle.json","state_url":"https://pith.science/pith/EOOFI2ZZUD6J4MIX3HT4HGHVIK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EOOFI2ZZUD6J4MIX3HT4HGHVIK/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":"2026-08-05T02:19:00Z","links":{"resolver":"https://pith.science/pith/EOOFI2ZZUD6J4MIX3HT4HGHVIK","bundle":"https://pith.science/pith/EOOFI2ZZUD6J4MIX3HT4HGHVIK/bundle.json","state":"https://pith.science/pith/EOOFI2ZZUD6J4MIX3HT4HGHVIK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EOOFI2ZZUD6J4MIX3HT4HGHVIK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:EOOFI2ZZUD6J4MIX3HT4HGHVIK","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":"84a098fe202487bb28814fb243451857fad4e17d153c9fcc38c22ef739b5f5ec","cross_cats_sorted":["math.CA","math.PR"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.FA","submitted_at":"2026-07-07T18:10:15Z","title_canon_sha256":"facaadb8135b0516f76efb40ce8789c81484056cc1e2b867b00f0418c9ae139e"},"schema_version":"1.0","source":{"id":"2607.06693","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.06693","created_at":"2026-07-09T00:19:23Z"},{"alias_kind":"arxiv_version","alias_value":"2607.06693v1","created_at":"2026-07-09T00:19:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.06693","created_at":"2026-07-09T00:19:23Z"},{"alias_kind":"pith_short_12","alias_value":"EOOFI2ZZUD6J","created_at":"2026-07-09T00:19:23Z"},{"alias_kind":"pith_short_16","alias_value":"EOOFI2ZZUD6J4MIX","created_at":"2026-07-09T00:19:23Z"},{"alias_kind":"pith_short_8","alias_value":"EOOFI2ZZ","created_at":"2026-07-09T00:19:23Z"}],"graph_snapshots":[{"event_id":"sha256:79fd6297dda8af192ff9a4dff0e55b40e2d024aa451ce8ebf171bbeb50dafa5d","target":"graph","created_at":"2026-07-09T00:19:23Z","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/2607.06693/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove that, after $L^2$ normalization, stable phase retrieval holds over the $L^2$-spans of independent real-valued centered random variables if and only if all but possibly one coordinate satisfies a uniform two-sided $L^1$ bound. This provides a complete characterization of stable phase retrieval for such subspaces, building upon the pioneering work of Calderbank--Daubechies--Freeman--Freeman and confirming the conjectured characterization communicated to us by those authors.\n  We provide two different proofs of this fact, both based on a decomposition of the $\\ell^2$-coefficients of each","authors_text":"Jaume de Dios Pont, Jo\\~ao P. G. Ramos, Mitchell A. Taylor, Pedro Abdalla","cross_cats":["math.CA","math.PR"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.FA","submitted_at":"2026-07-07T18:10:15Z","title":"Stable Phase Retrieval for Spans of Independent Random Variables"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.06693","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":"sha256:06eb2216d9dd13a12a5a3e70e28fb84b3d21fb929739b2b6b795082de06eed22","target":"record","created_at":"2026-07-09T00:19:23Z","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":"84a098fe202487bb28814fb243451857fad4e17d153c9fcc38c22ef739b5f5ec","cross_cats_sorted":["math.CA","math.PR"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.FA","submitted_at":"2026-07-07T18:10:15Z","title_canon_sha256":"facaadb8135b0516f76efb40ce8789c81484056cc1e2b867b00f0418c9ae139e"},"schema_version":"1.0","source":{"id":"2607.06693","kind":"arxiv","version":1}},"canonical_sha256":"239c546b39a0fc9e3117d9e7c398f542977c4a6d9895fe1eda779d15eefc451e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"239c546b39a0fc9e3117d9e7c398f542977c4a6d9895fe1eda779d15eefc451e","first_computed_at":"2026-07-09T00:19:23.687564Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-09T00:19:23.687564Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NEFA7zysL4cx288ffCMify94tat5klvcXtWbrtWETD3/bYHhIXkAEbSuOJ+EtgkvYoFyyVPW2ieSb9fWOQWUAw==","signature_status":"signed_v1","signed_at":"2026-07-09T00:19:23.688029Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.06693","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:06eb2216d9dd13a12a5a3e70e28fb84b3d21fb929739b2b6b795082de06eed22","sha256:79fd6297dda8af192ff9a4dff0e55b40e2d024aa451ce8ebf171bbeb50dafa5d"],"state_sha256":"ce7cd85fa839897912b543ac07f6295bc399c23513817ec3c16a8df767d1ee7b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/mhWDSQDGjcuAJUFpeBhQJneF6An5qQyj+ZdBLnXJXApu1LEVH5WYILF+8Z84JDEYldR/NKuwCsMwZTWc0OIDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-05T02:19:00.322656Z","bundle_sha256":"1846fa2b4de92b4de72a3aed0510c2430c3beca86675daab433b90c90c5b381d"}}