{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:HOFDDTZDXNYKJAJUINAVAR5G7X","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":"42f17b9a1f841d29d071bc1c5a693c249a4b18ec8ab3c67915238703ac10636e","cross_cats_sorted":["cs.DM","math-ph","math.CO","math.MP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2022-05-04T20:46:39Z","title_canon_sha256":"9a4f0fc29dad312e3a9b752c9b0ef47ccb93a99075b108fabeed9d9e60a4190b"},"schema_version":"1.0","source":{"id":"2205.02319","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2205.02319","created_at":"2026-07-05T06:39:29Z"},{"alias_kind":"arxiv_version","alias_value":"2205.02319v3","created_at":"2026-07-05T06:39:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2205.02319","created_at":"2026-07-05T06:39:29Z"},{"alias_kind":"pith_short_12","alias_value":"HOFDDTZDXNYK","created_at":"2026-07-05T06:39:29Z"},{"alias_kind":"pith_short_16","alias_value":"HOFDDTZDXNYKJAJU","created_at":"2026-07-05T06:39:29Z"},{"alias_kind":"pith_short_8","alias_value":"HOFDDTZD","created_at":"2026-07-05T06:39:29Z"}],"graph_snapshots":[{"event_id":"sha256:bb7934a9488e97fd968b07498eecaeca3482f4e49f819949e309a1e1b3889171","target":"graph","created_at":"2026-07-05T06:39:29Z","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/2205.02319/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the critical window of the symmetric binary perceptron, or equivalently, combinatorial discrepancy. Consider the problem of finding a binary vector $\\sigma$ satisfying $\\|A\\sigma\\|_\\infty \\le K$, where $A$ is an $\\alpha n \\times n$ matrix with iid Gaussian entries. For fixed $K$, at which densities $\\alpha$ is this constraint satisfaction problem (CSP) satisfiable? A sharp threshold was recently established by Perkins and Xu, and Abbe, Li, and Sly , answering this to first order. Namely, for each $K$ there exists an explicit critical density $\\alpha_c$ so that for any fixed $\\epsilon ","authors_text":"Dylan J. Altschuler","cross_cats":["cs.DM","math-ph","math.CO","math.MP"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2022-05-04T20:46:39Z","title":"Critical Window of The Symmetric Perceptron"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2205.02319","kind":"arxiv","version":3},"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":"sha256:90ae08be42eae2248be64a95d9ebe0f90611307a8de4a7ef49f55829d8fd47a1","target":"record","created_at":"2026-07-05T06:39:29Z","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":"42f17b9a1f841d29d071bc1c5a693c249a4b18ec8ab3c67915238703ac10636e","cross_cats_sorted":["cs.DM","math-ph","math.CO","math.MP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2022-05-04T20:46:39Z","title_canon_sha256":"9a4f0fc29dad312e3a9b752c9b0ef47ccb93a99075b108fabeed9d9e60a4190b"},"schema_version":"1.0","source":{"id":"2205.02319","kind":"arxiv","version":3}},"canonical_sha256":"3b8a31cf23bb70a4813443415047a6fdc93cb85b0dc9039fc118ad74280425a2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3b8a31cf23bb70a4813443415047a6fdc93cb85b0dc9039fc118ad74280425a2","first_computed_at":"2026-07-05T06:39:29.375281Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:39:29.375281Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fIs/o861OWAJsKzeaM33IF70EL2+8ri9oKky3Ciqd59qBNKn6CqLQ2TPEEGZ9JWI5jkqPU3hZym6sD6owynSCw==","signature_status":"signed_v1","signed_at":"2026-07-05T06:39:29.375816Z","signed_message":"canonical_sha256_bytes"},"source_id":"2205.02319","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:90ae08be42eae2248be64a95d9ebe0f90611307a8de4a7ef49f55829d8fd47a1","sha256:bb7934a9488e97fd968b07498eecaeca3482f4e49f819949e309a1e1b3889171"],"state_sha256":"b7ea7b53050e790850c762f9eb5351c20ce6572e3f0aae2c5d8b2f3ef3074687"}