{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:LPYTOKDBDH3YS3LH6RI3QRS4DD","short_pith_number":"pith:LPYTOKDB","canonical_record":{"source":{"id":"2508.03961","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-08-05T22:54:28Z","cross_cats_sorted":["cs.DM","cs.DS","math.PR"],"title_canon_sha256":"c2f6ae7521f04a925e2608fdcc17abcd26ea0b1224ca147cf51c43ab86bc787e","abstract_canon_sha256":"52219c706562ffe7723aaee0dce5d493fffd9a86a456f1513d6f62bbbe5f5b96"},"schema_version":"1.0"},"canonical_sha256":"5bf137286119f7896d67f451b8465c18c075f1c8465c6fdeceee0160a7f685fa","source":{"kind":"arxiv","id":"2508.03961","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2508.03961","created_at":"2026-07-05T12:07:40Z"},{"alias_kind":"arxiv_version","alias_value":"2508.03961v2","created_at":"2026-07-05T12:07:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2508.03961","created_at":"2026-07-05T12:07:40Z"},{"alias_kind":"pith_short_12","alias_value":"LPYTOKDBDH3Y","created_at":"2026-07-05T12:07:40Z"},{"alias_kind":"pith_short_16","alias_value":"LPYTOKDBDH3YS3LH","created_at":"2026-07-05T12:07:40Z"},{"alias_kind":"pith_short_8","alias_value":"LPYTOKDB","created_at":"2026-07-05T12:07:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:LPYTOKDBDH3YS3LH6RI3QRS4DD","target":"record","payload":{"canonical_record":{"source":{"id":"2508.03961","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-08-05T22:54:28Z","cross_cats_sorted":["cs.DM","cs.DS","math.PR"],"title_canon_sha256":"c2f6ae7521f04a925e2608fdcc17abcd26ea0b1224ca147cf51c43ab86bc787e","abstract_canon_sha256":"52219c706562ffe7723aaee0dce5d493fffd9a86a456f1513d6f62bbbe5f5b96"},"schema_version":"1.0"},"canonical_sha256":"5bf137286119f7896d67f451b8465c18c075f1c8465c6fdeceee0160a7f685fa","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:07:40.380696Z","signature_b64":"ynsHGSsgiLPQOWLuW+b79m3hTTFCUfHPQOsLublkbL8QSXrUIJYrUdsdg6OF980eo3lD90P0e2EGHgjhW2D2CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5bf137286119f7896d67f451b8465c18c075f1c8465c6fdeceee0160a7f685fa","last_reissued_at":"2026-07-05T12:07:40.380202Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:07:40.380202Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2508.03961","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T12:07:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oyuG+y8ffQbXvO2iTOSHIRDs4lYYaQQ80G2ciK9NiajBCIUC57hbmnqTslxNH4+lIbdqp3BVmKIIpRQnZageDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T11:10:26.052872Z"},"content_sha256":"1bf2981627441c6b84143fc19a6f6014a91988de97b58e9fd08c582758fe00ae","schema_version":"1.0","event_id":"sha256:1bf2981627441c6b84143fc19a6f6014a91988de97b58e9fd08c582758fe00ae"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:LPYTOKDBDH3YS3LH6RI3QRS4DD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Decoupling via Affine Spectral-Independence: Beck-Fiala and Koml\\'os Bounds Beyond Banaszczyk","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","cs.DS","math.PR"],"primary_cat":"math.CO","authors_text":"Haotian Jiang, Nikhil Bansal","submitted_at":"2025-08-05T22:54:28Z","abstract_excerpt":"The Beck-Fiala Conjecture [Discrete Appl. Math, 1981] asserts that any set system of $n$ elements with degree $k$ has combinatorial discrepancy $O(\\sqrt{k})$. A substantial generalization is the Koml\\'os Conjecture, which states that any $m \\times n$ matrix with unit length columns has discrepancy $O(1)$.\n  In this work, we resolve the Beck-Fiala Conjecture for $k \\geq \\log^2 n$. We also give an $\\widetilde{O}(\\sqrt{k} + \\sqrt{\\log n})$ bound for $k \\leq \\log^2 n$, where $\\widetilde{O}(\\cdot)$ hides $\\mathsf{poly}(\\log \\log n)$ factors. These bounds improve upon the $O(\\sqrt{k \\log n})$ bound "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.03961","kind":"arxiv","version":2},"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/2508.03961/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":"2026-07-05T12:07:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wzLCifYCGQadBW390Q9sPT/LMWfYli2BHc7Z+hBbiNPxHiDl7uQyHIZTza4dEE9kO5EA3/GwlMrK4GueRwN0Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T11:10:26.053403Z"},"content_sha256":"6cc67a2af067054fa14f1c67dcfed30a7a92da7a255408d2c91d2c75fbfd0a9f","schema_version":"1.0","event_id":"sha256:6cc67a2af067054fa14f1c67dcfed30a7a92da7a255408d2c91d2c75fbfd0a9f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LPYTOKDBDH3YS3LH6RI3QRS4DD/bundle.json","state_url":"https://pith.science/pith/LPYTOKDBDH3YS3LH6RI3QRS4DD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LPYTOKDBDH3YS3LH6RI3QRS4DD/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":"2026-08-20T11:10:26Z","links":{"resolver":"https://pith.science/pith/LPYTOKDBDH3YS3LH6RI3QRS4DD","bundle":"https://pith.science/pith/LPYTOKDBDH3YS3LH6RI3QRS4DD/bundle.json","state":"https://pith.science/pith/LPYTOKDBDH3YS3LH6RI3QRS4DD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LPYTOKDBDH3YS3LH6RI3QRS4DD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:LPYTOKDBDH3YS3LH6RI3QRS4DD","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":"52219c706562ffe7723aaee0dce5d493fffd9a86a456f1513d6f62bbbe5f5b96","cross_cats_sorted":["cs.DM","cs.DS","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-08-05T22:54:28Z","title_canon_sha256":"c2f6ae7521f04a925e2608fdcc17abcd26ea0b1224ca147cf51c43ab86bc787e"},"schema_version":"1.0","source":{"id":"2508.03961","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2508.03961","created_at":"2026-07-05T12:07:40Z"},{"alias_kind":"arxiv_version","alias_value":"2508.03961v2","created_at":"2026-07-05T12:07:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2508.03961","created_at":"2026-07-05T12:07:40Z"},{"alias_kind":"pith_short_12","alias_value":"LPYTOKDBDH3Y","created_at":"2026-07-05T12:07:40Z"},{"alias_kind":"pith_short_16","alias_value":"LPYTOKDBDH3YS3LH","created_at":"2026-07-05T12:07:40Z"},{"alias_kind":"pith_short_8","alias_value":"LPYTOKDB","created_at":"2026-07-05T12:07:40Z"}],"graph_snapshots":[{"event_id":"sha256:6cc67a2af067054fa14f1c67dcfed30a7a92da7a255408d2c91d2c75fbfd0a9f","target":"graph","created_at":"2026-07-05T12:07:40Z","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/2508.03961/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The Beck-Fiala Conjecture [Discrete Appl. Math, 1981] asserts that any set system of $n$ elements with degree $k$ has combinatorial discrepancy $O(\\sqrt{k})$. A substantial generalization is the Koml\\'os Conjecture, which states that any $m \\times n$ matrix with unit length columns has discrepancy $O(1)$.\n  In this work, we resolve the Beck-Fiala Conjecture for $k \\geq \\log^2 n$. We also give an $\\widetilde{O}(\\sqrt{k} + \\sqrt{\\log n})$ bound for $k \\leq \\log^2 n$, where $\\widetilde{O}(\\cdot)$ hides $\\mathsf{poly}(\\log \\log n)$ factors. These bounds improve upon the $O(\\sqrt{k \\log n})$ bound ","authors_text":"Haotian Jiang, Nikhil Bansal","cross_cats":["cs.DM","cs.DS","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-08-05T22:54:28Z","title":"Decoupling via Affine Spectral-Independence: Beck-Fiala and Koml\\'os Bounds Beyond Banaszczyk"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.03961","kind":"arxiv","version":2},"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":"sha256:1bf2981627441c6b84143fc19a6f6014a91988de97b58e9fd08c582758fe00ae","target":"record","created_at":"2026-07-05T12:07:40Z","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":"52219c706562ffe7723aaee0dce5d493fffd9a86a456f1513d6f62bbbe5f5b96","cross_cats_sorted":["cs.DM","cs.DS","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-08-05T22:54:28Z","title_canon_sha256":"c2f6ae7521f04a925e2608fdcc17abcd26ea0b1224ca147cf51c43ab86bc787e"},"schema_version":"1.0","source":{"id":"2508.03961","kind":"arxiv","version":2}},"canonical_sha256":"5bf137286119f7896d67f451b8465c18c075f1c8465c6fdeceee0160a7f685fa","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5bf137286119f7896d67f451b8465c18c075f1c8465c6fdeceee0160a7f685fa","first_computed_at":"2026-07-05T12:07:40.380202Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T12:07:40.380202Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ynsHGSsgiLPQOWLuW+b79m3hTTFCUfHPQOsLublkbL8QSXrUIJYrUdsdg6OF980eo3lD90P0e2EGHgjhW2D2CQ==","signature_status":"signed_v1","signed_at":"2026-07-05T12:07:40.380696Z","signed_message":"canonical_sha256_bytes"},"source_id":"2508.03961","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1bf2981627441c6b84143fc19a6f6014a91988de97b58e9fd08c582758fe00ae","sha256:6cc67a2af067054fa14f1c67dcfed30a7a92da7a255408d2c91d2c75fbfd0a9f"],"state_sha256":"7104a0b085c6cb98605c0d03a02f1f57b156b32777b78d533ca8a1a277ac7a5d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WxQ8BKa7EkPZjwqzieGi5rEZzfjD7NmSZ4ftUZPrfXiADtGtCaJ2hHWR3G+4vauScQ9WJYBymfxRlcYM20wKDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-20T11:10:26.058186Z","bundle_sha256":"56f4ee01d3f158bd79ade9feaded77821418d4d33118dee85a982223cfe6930c"}}