{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:AUVI4RBAFAI4RYW74JE5O447IO","short_pith_number":"pith:AUVI4RBA","canonical_record":{"source":{"id":"1608.07038","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2016-08-25T07:33:37Z","cross_cats_sorted":[],"title_canon_sha256":"36c640c0d8cb870cc47c79e24e0688e0a1e5686dc6e8307af83bcc333228918b","abstract_canon_sha256":"1fbb4c923a2191c697b52ff142934c31f57b8a4d5f7bd29515d58d0154a6b5e1"},"schema_version":"1.0"},"canonical_sha256":"052a8e44202811c8e2dfe249d7739f43b24dc3447ff60dabba4f4aa08607e33d","source":{"kind":"arxiv","id":"1608.07038","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.07038","created_at":"2026-05-18T00:55:57Z"},{"alias_kind":"arxiv_version","alias_value":"1608.07038v2","created_at":"2026-05-18T00:55:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.07038","created_at":"2026-05-18T00:55:57Z"},{"alias_kind":"pith_short_12","alias_value":"AUVI4RBAFAI4","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_16","alias_value":"AUVI4RBAFAI4RYW7","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_8","alias_value":"AUVI4RBA","created_at":"2026-05-18T12:30:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:AUVI4RBAFAI4RYW74JE5O447IO","target":"record","payload":{"canonical_record":{"source":{"id":"1608.07038","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2016-08-25T07:33:37Z","cross_cats_sorted":[],"title_canon_sha256":"36c640c0d8cb870cc47c79e24e0688e0a1e5686dc6e8307af83bcc333228918b","abstract_canon_sha256":"1fbb4c923a2191c697b52ff142934c31f57b8a4d5f7bd29515d58d0154a6b5e1"},"schema_version":"1.0"},"canonical_sha256":"052a8e44202811c8e2dfe249d7739f43b24dc3447ff60dabba4f4aa08607e33d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:55:57.715293Z","signature_b64":"5cbOatl3g34tB/ZcpeUsYVGCu4Wv2C0yB+rWMYRL5USh5Hunm+V4cDk+wvmZN2RhAj416W5ULszFUmQKuzEBDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"052a8e44202811c8e2dfe249d7739f43b24dc3447ff60dabba4f4aa08607e33d","last_reissued_at":"2026-05-18T00:55:57.714747Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:55:57.714747Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1608.07038","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":"2026-05-18T00:55:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xDGGYV4zkrgKXm7NiEoT7TACu5JXcZ+Fn0qPHVyDCzJLzuv3duNQKOKMIXySuGTZFOc1qouGNDlCPwDKLbwnDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T18:41:08.147994Z"},"content_sha256":"fc935b05e210be8f051db32ff9fa4274a47e41c7fef2f1fb59ed4bf03677a9c6","schema_version":"1.0","event_id":"sha256:fc935b05e210be8f051db32ff9fa4274a47e41c7fef2f1fb59ed4bf03677a9c6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:AUVI4RBAFAI4RYW74JE5O447IO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The (minimum) rank of typical fooling set matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DM","authors_text":"Dirk Oliver Theis, Mozhgan Pourmoradnasseri","submitted_at":"2016-08-25T07:33:37Z","abstract_excerpt":"A fooling-set matrix has nonzero diagonal, but at least one in every pair of diagonally opposite entries is 0. Dietzfelbinger et al. '96 proved that the rank of such a matrix is at least $\\sqrt n$. It is known that the bound is tight (up to a multiplicative constant).\n  We ask for the \"typical\" minimum rank of a fooling-set matrix: For a fooling-set zero-nonzero pattern chosen at random, is the minimum rank of a matrix with that zero-nonzero pattern over a field $\\mathbb F$ closer to its lower bound $\\sqrt{n}$ or to its upper bound $n$? We study random patterns with a given density $p$, and pr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.07038","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":""},"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":"2026-05-18T00:55:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9qdQop5jTWEJ2WKa2r7RLZ/hqWiLrwcjqZVudYP5RSapZHttNXh0NOpi+yrzTToyFkuc2wHwxwYTk7R688QoDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T18:41:08.148353Z"},"content_sha256":"91a5be6ddf6cfa26cb243fd38fb1dcd1d7d638c3f7673c3bb60748be713f054c","schema_version":"1.0","event_id":"sha256:91a5be6ddf6cfa26cb243fd38fb1dcd1d7d638c3f7673c3bb60748be713f054c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AUVI4RBAFAI4RYW74JE5O447IO/bundle.json","state_url":"https://pith.science/pith/AUVI4RBAFAI4RYW74JE5O447IO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AUVI4RBAFAI4RYW74JE5O447IO/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":"2026-06-03T18:41:08Z","links":{"resolver":"https://pith.science/pith/AUVI4RBAFAI4RYW74JE5O447IO","bundle":"https://pith.science/pith/AUVI4RBAFAI4RYW74JE5O447IO/bundle.json","state":"https://pith.science/pith/AUVI4RBAFAI4RYW74JE5O447IO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AUVI4RBAFAI4RYW74JE5O447IO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:AUVI4RBAFAI4RYW74JE5O447IO","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":"1fbb4c923a2191c697b52ff142934c31f57b8a4d5f7bd29515d58d0154a6b5e1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2016-08-25T07:33:37Z","title_canon_sha256":"36c640c0d8cb870cc47c79e24e0688e0a1e5686dc6e8307af83bcc333228918b"},"schema_version":"1.0","source":{"id":"1608.07038","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.07038","created_at":"2026-05-18T00:55:57Z"},{"alias_kind":"arxiv_version","alias_value":"1608.07038v2","created_at":"2026-05-18T00:55:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.07038","created_at":"2026-05-18T00:55:57Z"},{"alias_kind":"pith_short_12","alias_value":"AUVI4RBAFAI4","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_16","alias_value":"AUVI4RBAFAI4RYW7","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_8","alias_value":"AUVI4RBA","created_at":"2026-05-18T12:30:07Z"}],"graph_snapshots":[{"event_id":"sha256:91a5be6ddf6cfa26cb243fd38fb1dcd1d7d638c3f7673c3bb60748be713f054c","target":"graph","created_at":"2026-05-18T00:55:57Z","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"},"paper":{"abstract_excerpt":"A fooling-set matrix has nonzero diagonal, but at least one in every pair of diagonally opposite entries is 0. Dietzfelbinger et al. '96 proved that the rank of such a matrix is at least $\\sqrt n$. It is known that the bound is tight (up to a multiplicative constant).\n  We ask for the \"typical\" minimum rank of a fooling-set matrix: For a fooling-set zero-nonzero pattern chosen at random, is the minimum rank of a matrix with that zero-nonzero pattern over a field $\\mathbb F$ closer to its lower bound $\\sqrt{n}$ or to its upper bound $n$? We study random patterns with a given density $p$, and pr","authors_text":"Dirk Oliver Theis, Mozhgan Pourmoradnasseri","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2016-08-25T07:33:37Z","title":"The (minimum) rank of typical fooling set matrices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.07038","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:fc935b05e210be8f051db32ff9fa4274a47e41c7fef2f1fb59ed4bf03677a9c6","target":"record","created_at":"2026-05-18T00:55:57Z","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":"1fbb4c923a2191c697b52ff142934c31f57b8a4d5f7bd29515d58d0154a6b5e1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2016-08-25T07:33:37Z","title_canon_sha256":"36c640c0d8cb870cc47c79e24e0688e0a1e5686dc6e8307af83bcc333228918b"},"schema_version":"1.0","source":{"id":"1608.07038","kind":"arxiv","version":2}},"canonical_sha256":"052a8e44202811c8e2dfe249d7739f43b24dc3447ff60dabba4f4aa08607e33d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"052a8e44202811c8e2dfe249d7739f43b24dc3447ff60dabba4f4aa08607e33d","first_computed_at":"2026-05-18T00:55:57.714747Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:55:57.714747Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5cbOatl3g34tB/ZcpeUsYVGCu4Wv2C0yB+rWMYRL5USh5Hunm+V4cDk+wvmZN2RhAj416W5ULszFUmQKuzEBDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:55:57.715293Z","signed_message":"canonical_sha256_bytes"},"source_id":"1608.07038","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fc935b05e210be8f051db32ff9fa4274a47e41c7fef2f1fb59ed4bf03677a9c6","sha256:91a5be6ddf6cfa26cb243fd38fb1dcd1d7d638c3f7673c3bb60748be713f054c"],"state_sha256":"eac6fe292247de88a1d8b741ccfc889516128adb82b0e7304622b275dd976c20"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gVjXcCytoWrnkSna6ZWJGXjoQ8b31oLOqLM83DSRMdCgpU2KhaZySVa3wD7sHiTJ/UbprKj44kLUAeMaO8d9Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T18:41:08.150301Z","bundle_sha256":"8001021f7d26f94bfba900de31f31b1f59bedf278fccaf01f4d61ced8e3840a4"}}