{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:4OUATUQJSSZGVP5ONDAARU5HZT","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":"91de74cb41f9111b48bc6dc8a269b82aeb344bd99cf43cd62eec3f0596e52164","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2026-08-06T17:46:47Z","title_canon_sha256":"2abd81dc98c3851b2294b964f8c5f6d650bf393c194f5b2d0100e420d1f49fb8"},"schema_version":"1.0","source":{"id":"2608.08238","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.08238","created_at":"2026-08-11T01:22:05Z"},{"alias_kind":"arxiv_version","alias_value":"2608.08238v1","created_at":"2026-08-11T01:22:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.08238","created_at":"2026-08-11T01:22:05Z"},{"alias_kind":"pith_short_12","alias_value":"4OUATUQJSSZG","created_at":"2026-08-11T01:22:05Z"},{"alias_kind":"pith_short_16","alias_value":"4OUATUQJSSZGVP5O","created_at":"2026-08-11T01:22:05Z"},{"alias_kind":"pith_short_8","alias_value":"4OUATUQJ","created_at":"2026-08-11T01:22:05Z"}],"graph_snapshots":[{"event_id":"sha256:0fd5685e369a1388fe19c3d98daa12b20fc68fe1176568cfb9074148cc2aee20","target":"graph","created_at":"2026-08-11T01:22:05Z","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/2608.08238/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For the $n\\times n$ lower-triangular all-ones matrix $Q$, we prove a near-optimal lower bound \\[ \\gamma_{2,1}(Q) := \\inf_{Q=AB} \\|A\\|_{2\\to\\infty}\\|B\\|_{1\\to1} =\n\\Omega\\!\\left( \\frac{\\log^{3/2}n}{(\\log\\log n)^{3/2}} \\right), \\] where the infimum ranges over real factorizations of arbitrary finite inner dimension. This cost is a central parameter in space bounds for factorization-based rank and quantile estimation in turnstile streams and in error bounds for matrix mechanisms for continual counting under pure differential privacy. The proof combines right-sided Haar projections with a scale-dep","authors_text":"David P. Woodruff, Honghao Lin, Vahab Mirrokni","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2026-08-06T17:46:47Z","title":"A Near-Optimal Lower Bound for Prefix-Matrix Factorizations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.08238","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:ac853b1fb731bf7ffcee5001dd7fff6b7e41b3fc8ce81f50a48782df278cb106","target":"record","created_at":"2026-08-11T01:22:05Z","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":"91de74cb41f9111b48bc6dc8a269b82aeb344bd99cf43cd62eec3f0596e52164","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2026-08-06T17:46:47Z","title_canon_sha256":"2abd81dc98c3851b2294b964f8c5f6d650bf393c194f5b2d0100e420d1f49fb8"},"schema_version":"1.0","source":{"id":"2608.08238","kind":"arxiv","version":1}},"canonical_sha256":"e3a809d20994b26abfae68c008d3a7ccc6f54f7cc99003fb986137014aba7e25","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e3a809d20994b26abfae68c008d3a7ccc6f54f7cc99003fb986137014aba7e25","first_computed_at":"2026-08-11T01:22:05.850993Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-11T01:22:05.850993Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zX1+O4Y5xMnHJaRFVo8Q+BO22RLI4AK7DzUHhZWGHlJ36FL2DozSh8SxO2URnfxYKliyPoE2PqncyUAFkCkMBQ==","signature_status":"signed_v1","signed_at":"2026-08-11T01:22:05.853451Z","signed_message":"canonical_sha256_bytes"},"source_id":"2608.08238","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ac853b1fb731bf7ffcee5001dd7fff6b7e41b3fc8ce81f50a48782df278cb106","sha256:0fd5685e369a1388fe19c3d98daa12b20fc68fe1176568cfb9074148cc2aee20"],"state_sha256":"77be2b9319b16d76c026c94b81700a5d4e095efef8efbf4c004cc5b73cf41f94"}