{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:FXOE74RNILRMHESQ3H3WFNPYUY","short_pith_number":"pith:FXOE74RN","schema_version":"1.0","canonical_sha256":"2ddc4ff22d42e2c39250d9f762b5f8a6276743a8f29ba97631aa5b48d251e426","source":{"kind":"arxiv","id":"2605.18335","version":1},"attestation_state":"computed","paper":{"title":"A Note on Second-Order Expected Maximum-Load Bounds for Binary Linear Hashing","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Binary linear hashing matches fully independent hashing on the second-order term in expected maximum load.","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Nader H. Bshouty","submitted_at":"2026-05-18T12:51:10Z","abstract_excerpt":"Let $S\\subseteq F_2^u$ have size $n=2^\\ell$, and let $h:F_2^u\\to F_2^\\ell$ be a uniformly random linear map. For $y\\in F_2^\\ell$, write $Load_h(y):=|h^{-1}(y)\\cap S|$, and let $M(S,h):=\\max_{y\\in F_2^\\ell} Load_h(y)$ be the maximum load. Jaber, Kumar and Zuckerman (STOC 2025) proved that the expected maximum load of $h$ on $S$ is at most $16\\log n/\\log\\log n$, matching the fully independent keys-into-bins scale up to constants. Their proof also gives the tail estimate \\[\n  \\Pr\\left[\n  M(S,h)\\ge R\\frac{\\log n}{\\log\\log n}\n  \\right]\n  \\le O\\left(\\frac{1}{R^{2}}\\right). \\] We record a base optimi"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":true,"formal_links_present":true},"canonical_record":{"source":{"id":"2605.18335","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DS","submitted_at":"2026-05-18T12:51:10Z","cross_cats_sorted":[],"title_canon_sha256":"3887c5235e275e953a17b80fd67cc83fd2c3d0e96964d032061ffc221b871f44","abstract_canon_sha256":"db429c0c8b53c9eaf56c9cb1e1c2d7336cd6046192fbae2e8edfc4acb0bf184c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:05:55.768732Z","signature_b64":"qZiv2YnBuV20IXjPc8kvHBeogr6w55gg2KHAVxTu8xcUwZC/I5+FcQVU57uie1BvuNKFOEZhuawHUmxxJiyCBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2ddc4ff22d42e2c39250d9f762b5f8a6276743a8f29ba97631aa5b48d251e426","last_reissued_at":"2026-05-20T00:05:55.767935Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:05:55.767935Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Note on Second-Order Expected Maximum-Load Bounds for Binary Linear Hashing","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Binary linear hashing matches fully independent hashing on the second-order term in expected maximum load.","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Nader H. Bshouty","submitted_at":"2026-05-18T12:51:10Z","abstract_excerpt":"Let $S\\subseteq F_2^u$ have size $n=2^\\ell$, and let $h:F_2^u\\to F_2^\\ell$ be a uniformly random linear map. For $y\\in F_2^\\ell$, write $Load_h(y):=|h^{-1}(y)\\cap S|$, and let $M(S,h):=\\max_{y\\in F_2^\\ell} Load_h(y)$ be the maximum load. Jaber, Kumar and Zuckerman (STOC 2025) proved that the expected maximum load of $h$ on $S$ is at most $16\\log n/\\log\\log n$, matching the fully independent keys-into-bins scale up to constants. Their proof also gives the tail estimate \\[\n  \\Pr\\left[\n  M(S,h)\\ge R\\frac{\\log n}{\\log\\log n}\n  \\right]\n  \\le O\\left(\\frac{1}{R^{2}}\\right). \\] We record a base optimi"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For every R>1 satisfying R ℓ^{1-1/R} ≥ D ln ℓ, Pr[M(S,h) ≥ R log n / log log n] ≤ O( (log log n)^2 / (R^2 (log n)^{2-2/R}) ), which integrates to E[M(S,h)] ≤ (1 + (1+o(1)) log log log n / log log n) * log n / log log n.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The base optimization of the exponential-potential function in the proof from Jaber et al. (STOC 2025) can be carried through without introducing new error terms that invalidate the improved exponent 2-2/R in the tail bound (see the derivation of the tail estimate in the note).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Binary linear hashing matches fully independent hashing in the leading term and dominant second-order correction of expected maximum load up to a 1+o(1) factor.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Binary linear hashing matches fully independent hashing on the second-order term in expected maximum load.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"ed779228ebec8b7b2cfe01f3709aeed2449161c70d29dab34ce5df0b0a92b31d"},"source":{"id":"2605.18335","kind":"arxiv","version":1},"verdict":{"id":"f129dc42-8e3e-44af-bcde-f84d4a9666b3","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T23:47:58.962125Z","strongest_claim":"For every R>1 satisfying R ℓ^{1-1/R} ≥ D ln ℓ, Pr[M(S,h) ≥ R log n / log log n] ≤ O( (log log n)^2 / (R^2 (log n)^{2-2/R}) ), which integrates to E[M(S,h)] ≤ (1 + (1+o(1)) log log log n / log log n) * log n / log log n.","one_line_summary":"Binary linear hashing matches fully independent hashing in the leading term and dominant second-order correction of expected maximum load up to a 1+o(1) factor.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The base optimization of the exponential-potential function in the proof from Jaber et al. (STOC 2025) can be carried through without introducing new error terms that invalidate the improved exponent 2-2/R in the tail bound (see the derivation of the tail estimate in the note).","pith_extraction_headline":"Binary linear hashing matches fully independent hashing on the second-order term in expected maximum load."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.18335/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-20T00:02:35.073119Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-20T00:01:20.436118Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T23:33:35.170964Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T23:21:58.838630Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"83b630963c9508869683cb583c72e3584ddd87cc63d61a0544dde548a4815be0"},"references":{"count":12,"sample":[{"doi":"","year":1997,"title":"Is linear hashing good? InProceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, pages 465–474, 1997","work_id":"24ef6bfd-aa5d-4754-b830-195d111fb2fd","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1999,"title":"Linear hash functions.Journal of the ACM, 46(5):667–683, 1999","work_id":"d776c225-f8ef-45ea-a47f-84f863478abe","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Kumar, and David Zuckerman","work_id":"20173400-cf66-48f1-97ff-907b5b3af378","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Also available as arXiv:2505.14061","work_id":"56c62d82-14d1-4b73-b9fd-bc344172a536","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1998,"title":"Balls into bins","work_id":"257c7425-0b23-47fa-8569-94696123dc91","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":12,"snapshot_sha256":"66585bb495ba56411ae4703f55be75bd7040ed45bebbc1ed4b2cab7dd60a3867","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"582406f642fc484a6539a8677565c35ab17429cd0cd50b2a0c4b762999dc3f0d"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2605.18335","created_at":"2026-05-20T00:05:55.768074+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.18335v1","created_at":"2026-05-20T00:05:55.768074+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.18335","created_at":"2026-05-20T00:05:55.768074+00:00"},{"alias_kind":"pith_short_12","alias_value":"FXOE74RNILRM","created_at":"2026-05-20T00:05:55.768074+00:00"},{"alias_kind":"pith_short_16","alias_value":"FXOE74RNILRMHESQ","created_at":"2026-05-20T00:05:55.768074+00:00"},{"alias_kind":"pith_short_8","alias_value":"FXOE74RN","created_at":"2026-05-20T00:05:55.768074+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":2,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/FXOE74RNILRMHESQ3H3WFNPYUY","json":"https://pith.science/pith/FXOE74RNILRMHESQ3H3WFNPYUY.json","graph_json":"https://pith.science/api/pith-number/FXOE74RNILRMHESQ3H3WFNPYUY/graph.json","events_json":"https://pith.science/api/pith-number/FXOE74RNILRMHESQ3H3WFNPYUY/events.json","paper":"https://pith.science/paper/FXOE74RN"},"agent_actions":{"view_html":"https://pith.science/pith/FXOE74RNILRMHESQ3H3WFNPYUY","download_json":"https://pith.science/pith/FXOE74RNILRMHESQ3H3WFNPYUY.json","view_paper":"https://pith.science/paper/FXOE74RN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.18335&json=true","fetch_graph":"https://pith.science/api/pith-number/FXOE74RNILRMHESQ3H3WFNPYUY/graph.json","fetch_events":"https://pith.science/api/pith-number/FXOE74RNILRMHESQ3H3WFNPYUY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FXOE74RNILRMHESQ3H3WFNPYUY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FXOE74RNILRMHESQ3H3WFNPYUY/action/storage_attestation","attest_author":"https://pith.science/pith/FXOE74RNILRMHESQ3H3WFNPYUY/action/author_attestation","sign_citation":"https://pith.science/pith/FXOE74RNILRMHESQ3H3WFNPYUY/action/citation_signature","submit_replication":"https://pith.science/pith/FXOE74RNILRMHESQ3H3WFNPYUY/action/replication_record"}},"created_at":"2026-05-20T00:05:55.768074+00:00","updated_at":"2026-05-20T00:05:55.768074+00:00"}