{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:B4JNFI6W4DHIYWIEIE2WBGVVTR","short_pith_number":"pith:B4JNFI6W","canonical_record":{"source":{"id":"1908.11255","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2019-08-29T14:26:03Z","cross_cats_sorted":["cs.NA","math.CO","math.NA"],"title_canon_sha256":"84508eae5aca3f0a2099193b342566f781485c25c91eba96d69522cc38d390de","abstract_canon_sha256":"f3b9d638da46c0b3e249b59e99a34194213697edd78aab88638d7121f20f4ef3"},"schema_version":"1.0"},"canonical_sha256":"0f12d2a3d6e0ce8c59044135609ab59c442e979da2e6c710e80ad88cb2892356","source":{"kind":"arxiv","id":"1908.11255","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.11255","created_at":"2026-07-05T03:11:09Z"},{"alias_kind":"arxiv_version","alias_value":"1908.11255v3","created_at":"2026-07-05T03:11:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.11255","created_at":"2026-07-05T03:11:09Z"},{"alias_kind":"pith_short_12","alias_value":"B4JNFI6W4DHI","created_at":"2026-07-05T03:11:09Z"},{"alias_kind":"pith_short_16","alias_value":"B4JNFI6W4DHIYWIE","created_at":"2026-07-05T03:11:09Z"},{"alias_kind":"pith_short_8","alias_value":"B4JNFI6W","created_at":"2026-07-05T03:11:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:B4JNFI6W4DHIYWIEIE2WBGVVTR","target":"record","payload":{"canonical_record":{"source":{"id":"1908.11255","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2019-08-29T14:26:03Z","cross_cats_sorted":["cs.NA","math.CO","math.NA"],"title_canon_sha256":"84508eae5aca3f0a2099193b342566f781485c25c91eba96d69522cc38d390de","abstract_canon_sha256":"f3b9d638da46c0b3e249b59e99a34194213697edd78aab88638d7121f20f4ef3"},"schema_version":"1.0"},"canonical_sha256":"0f12d2a3d6e0ce8c59044135609ab59c442e979da2e6c710e80ad88cb2892356","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:11:09.393715Z","signature_b64":"xQfx8rj2+Lppni+4Ybw2YJ4+yGZ0uCk1Ku2zrFNmWJCbHAnVtzdyO86Pq0AiT7/NIo4VpXqUhotXCzZMO09pBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0f12d2a3d6e0ce8c59044135609ab59c442e979da2e6c710e80ad88cb2892356","last_reissued_at":"2026-07-05T03:11:09.393240Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:11:09.393240Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1908.11255","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T03:11:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2jSrb3HSkvjsvo8g8pxeY4Ww0UNY1Aygx+hoBRQW9JcPFpAPINWjkqdNP25JBPxhE9D9VYxvSsfOQ8qY/HbzAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T00:12:43.827990Z"},"content_sha256":"e4ba5c0d190a33ddd2355e9a06a816d2ad2bc54a86581b1a2752b4dda054416d","schema_version":"1.0","event_id":"sha256:e4ba5c0d190a33ddd2355e9a06a816d2ad2bc54a86581b1a2752b4dda054416d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:B4JNFI6W4DHIYWIEIE2WBGVVTR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Quantitative invertibility of random matrices: a combinatorial perspective","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.NA","math.CO","math.NA"],"primary_cat":"math.PR","authors_text":"Vishesh Jain","submitted_at":"2019-08-29T14:26:03Z","abstract_excerpt":"We study the lower tail behavior of the least singular value of an $n\\times n$ random matrix $M_n := M+N_n$, where $M$ is a fixed complex matrix with operator norm at most $\\exp(n^{c})$ and $N_n$ is a random matrix, each of whose entries is an independent copy of a complex random variable with mean $0$ and variance $1$. Motivated by applications, our focus is on obtaining bounds which hold with extremely high probability, rather than on the least singular value of a typical such matrix.\n  This setting has previously been considered in a series of influential works by Tao and Vu, most notably i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.11255","kind":"arxiv","version":3},"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/1908.11255/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":"2026-07-05T03:11:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"F+2k2PnPonkmqVZgoR0MOHr6ifmZQjp7Yvupkt0y553uWfeakaLD5xXN4hegsIUqlAovQqbWSyTw4nx9XTapBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T00:12:43.828554Z"},"content_sha256":"b3282e14a026314579052d09d1d62a1d06ed756143db3e85a5157200671c7554","schema_version":"1.0","event_id":"sha256:b3282e14a026314579052d09d1d62a1d06ed756143db3e85a5157200671c7554"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/B4JNFI6W4DHIYWIEIE2WBGVVTR/bundle.json","state_url":"https://pith.science/pith/B4JNFI6W4DHIYWIEIE2WBGVVTR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/B4JNFI6W4DHIYWIEIE2WBGVVTR/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":"2026-08-19T00:12:43Z","links":{"resolver":"https://pith.science/pith/B4JNFI6W4DHIYWIEIE2WBGVVTR","bundle":"https://pith.science/pith/B4JNFI6W4DHIYWIEIE2WBGVVTR/bundle.json","state":"https://pith.science/pith/B4JNFI6W4DHIYWIEIE2WBGVVTR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/B4JNFI6W4DHIYWIEIE2WBGVVTR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:B4JNFI6W4DHIYWIEIE2WBGVVTR","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":"f3b9d638da46c0b3e249b59e99a34194213697edd78aab88638d7121f20f4ef3","cross_cats_sorted":["cs.NA","math.CO","math.NA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2019-08-29T14:26:03Z","title_canon_sha256":"84508eae5aca3f0a2099193b342566f781485c25c91eba96d69522cc38d390de"},"schema_version":"1.0","source":{"id":"1908.11255","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.11255","created_at":"2026-07-05T03:11:09Z"},{"alias_kind":"arxiv_version","alias_value":"1908.11255v3","created_at":"2026-07-05T03:11:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.11255","created_at":"2026-07-05T03:11:09Z"},{"alias_kind":"pith_short_12","alias_value":"B4JNFI6W4DHI","created_at":"2026-07-05T03:11:09Z"},{"alias_kind":"pith_short_16","alias_value":"B4JNFI6W4DHIYWIE","created_at":"2026-07-05T03:11:09Z"},{"alias_kind":"pith_short_8","alias_value":"B4JNFI6W","created_at":"2026-07-05T03:11:09Z"}],"graph_snapshots":[{"event_id":"sha256:b3282e14a026314579052d09d1d62a1d06ed756143db3e85a5157200671c7554","target":"graph","created_at":"2026-07-05T03:11:09Z","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/1908.11255/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the lower tail behavior of the least singular value of an $n\\times n$ random matrix $M_n := M+N_n$, where $M$ is a fixed complex matrix with operator norm at most $\\exp(n^{c})$ and $N_n$ is a random matrix, each of whose entries is an independent copy of a complex random variable with mean $0$ and variance $1$. Motivated by applications, our focus is on obtaining bounds which hold with extremely high probability, rather than on the least singular value of a typical such matrix.\n  This setting has previously been considered in a series of influential works by Tao and Vu, most notably i","authors_text":"Vishesh Jain","cross_cats":["cs.NA","math.CO","math.NA"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2019-08-29T14:26:03Z","title":"Quantitative invertibility of random matrices: a combinatorial perspective"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.11255","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:e4ba5c0d190a33ddd2355e9a06a816d2ad2bc54a86581b1a2752b4dda054416d","target":"record","created_at":"2026-07-05T03:11:09Z","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":"f3b9d638da46c0b3e249b59e99a34194213697edd78aab88638d7121f20f4ef3","cross_cats_sorted":["cs.NA","math.CO","math.NA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2019-08-29T14:26:03Z","title_canon_sha256":"84508eae5aca3f0a2099193b342566f781485c25c91eba96d69522cc38d390de"},"schema_version":"1.0","source":{"id":"1908.11255","kind":"arxiv","version":3}},"canonical_sha256":"0f12d2a3d6e0ce8c59044135609ab59c442e979da2e6c710e80ad88cb2892356","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0f12d2a3d6e0ce8c59044135609ab59c442e979da2e6c710e80ad88cb2892356","first_computed_at":"2026-07-05T03:11:09.393240Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:11:09.393240Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xQfx8rj2+Lppni+4Ybw2YJ4+yGZ0uCk1Ku2zrFNmWJCbHAnVtzdyO86Pq0AiT7/NIo4VpXqUhotXCzZMO09pBA==","signature_status":"signed_v1","signed_at":"2026-07-05T03:11:09.393715Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.11255","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e4ba5c0d190a33ddd2355e9a06a816d2ad2bc54a86581b1a2752b4dda054416d","sha256:b3282e14a026314579052d09d1d62a1d06ed756143db3e85a5157200671c7554"],"state_sha256":"a8a723d37291cd3ace01309fda30a82b6d73a5e8a312a793c2e1f9126bc1078e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZDbp6bifMqMRfNybITWaiUS7Xvt8+07gQsKyfmQj3WEJ0/+0pB9R94zOop2X7fQiC80ew7dEYJqbYpb0zpV9CA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-19T00:12:43.834259Z","bundle_sha256":"895e26eeda52df03c88e7267dce3ac3e95d5cb2dcdacaa5251e0a6a0165d2474"}}