{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:CKWEZZ2OQXQR52JQTV3IULC6PL","short_pith_number":"pith:CKWEZZ2O","canonical_record":{"source":{"id":"2603.04365","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-03-04T18:30:04Z","cross_cats_sorted":["cs.NA","math.NA","math.ST","stat.TH"],"title_canon_sha256":"eee4193c823f266b2c8e677e2b42585b4c42badb8cbdef8d0dd4ac84b4b452ca","abstract_canon_sha256":"630ea065d2fe48c86f9310fdaa0af49a287f7eb42a2797994f898d8d35f4b377"},"schema_version":"1.0"},"canonical_sha256":"12ac4ce74e85e11ee9309d768a2c5e7ad6e9d4061c2804cc1bd21f5df4178a5a","source":{"kind":"arxiv","id":"2603.04365","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2603.04365","created_at":"2026-07-07T02:18:36Z"},{"alias_kind":"arxiv_version","alias_value":"2603.04365v3","created_at":"2026-07-07T02:18:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2603.04365","created_at":"2026-07-07T02:18:36Z"},{"alias_kind":"pith_short_12","alias_value":"CKWEZZ2OQXQR","created_at":"2026-07-07T02:18:36Z"},{"alias_kind":"pith_short_16","alias_value":"CKWEZZ2OQXQR52JQ","created_at":"2026-07-07T02:18:36Z"},{"alias_kind":"pith_short_8","alias_value":"CKWEZZ2O","created_at":"2026-07-07T02:18:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:CKWEZZ2OQXQR52JQTV3IULC6PL","target":"record","payload":{"canonical_record":{"source":{"id":"2603.04365","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-03-04T18:30:04Z","cross_cats_sorted":["cs.NA","math.NA","math.ST","stat.TH"],"title_canon_sha256":"eee4193c823f266b2c8e677e2b42585b4c42badb8cbdef8d0dd4ac84b4b452ca","abstract_canon_sha256":"630ea065d2fe48c86f9310fdaa0af49a287f7eb42a2797994f898d8d35f4b377"},"schema_version":"1.0"},"canonical_sha256":"12ac4ce74e85e11ee9309d768a2c5e7ad6e9d4061c2804cc1bd21f5df4178a5a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-07T02:18:36.405790Z","signature_b64":"TjQ2mjnku+Ku3VzDNuDw6BZ/kw+pH+v7KEIcLLhPgkga7sOuwHRds5WU7EDItiyeuTrJYWNdZDU60QwWm/VbAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"12ac4ce74e85e11ee9309d768a2c5e7ad6e9d4061c2804cc1bd21f5df4178a5a","last_reissued_at":"2026-07-07T02:18:36.404712Z","signature_status":"signed_v1","first_computed_at":"2026-07-07T02:18:36.404712Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2603.04365","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":"2026-07-07T02:18:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZzQd2xmwwOaCarEMkcqsTTPLNnAxlhRbRTvdvqc5TEw2pkY4cDKFV8OoAnoDzKI+PUlbJvpV/Nrut13lyGh+Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T12:12:59.957677Z"},"content_sha256":"24670cb7aaaaa962b89e313f726173d0141183713df6338212f1a639d991c581","schema_version":"1.0","event_id":"sha256:24670cb7aaaaa962b89e313f726173d0141183713df6338212f1a639d991c581"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:CKWEZZ2OQXQR52JQTV3IULC6PL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Comparison theorems for the extreme eigenvalues of a random symmetric matrix","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA","math.NA","math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Joel A. Tropp","submitted_at":"2026-03-04T18:30:04Z","abstract_excerpt":"This paper establishes a comparison theorem for the maximum eigenvalue of a sum of independent random symmetric matrices. The theorem states that the maximum eigenvalue of the matrix sum is dominated by the maximum eigenvalue of a Gaussian random matrix whose statistics match the sum, and it strengthens previous results of this type. Corollaries address the minimum eigenvalue and the spectral norm; the proof strategy also extends to matrix martingale sequences.\n  The comparison methodology is powerful because of the vast arsenal of tools for treating Gaussian random matrices. As applications, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2603.04365","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/2603.04365/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":"2026-07-07T02:18:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5Mb4ezmSK3HUCweWQhm+bGx6pVdI+U9LetAlodsgJPzTixKdFhmbDoJtOlyu+P2OgiJCKUaBFGufqdZsdRatDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T12:12:59.958584Z"},"content_sha256":"d227c3d0f11d474d03bb89e161dd3c70ee242a54b6e25231a34f2fd081813f93","schema_version":"1.0","event_id":"sha256:d227c3d0f11d474d03bb89e161dd3c70ee242a54b6e25231a34f2fd081813f93"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CKWEZZ2OQXQR52JQTV3IULC6PL/bundle.json","state_url":"https://pith.science/pith/CKWEZZ2OQXQR52JQTV3IULC6PL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CKWEZZ2OQXQR52JQTV3IULC6PL/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":"2026-08-06T12:12:59Z","links":{"resolver":"https://pith.science/pith/CKWEZZ2OQXQR52JQTV3IULC6PL","bundle":"https://pith.science/pith/CKWEZZ2OQXQR52JQTV3IULC6PL/bundle.json","state":"https://pith.science/pith/CKWEZZ2OQXQR52JQTV3IULC6PL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CKWEZZ2OQXQR52JQTV3IULC6PL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:CKWEZZ2OQXQR52JQTV3IULC6PL","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":"630ea065d2fe48c86f9310fdaa0af49a287f7eb42a2797994f898d8d35f4b377","cross_cats_sorted":["cs.NA","math.NA","math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-03-04T18:30:04Z","title_canon_sha256":"eee4193c823f266b2c8e677e2b42585b4c42badb8cbdef8d0dd4ac84b4b452ca"},"schema_version":"1.0","source":{"id":"2603.04365","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2603.04365","created_at":"2026-07-07T02:18:36Z"},{"alias_kind":"arxiv_version","alias_value":"2603.04365v3","created_at":"2026-07-07T02:18:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2603.04365","created_at":"2026-07-07T02:18:36Z"},{"alias_kind":"pith_short_12","alias_value":"CKWEZZ2OQXQR","created_at":"2026-07-07T02:18:36Z"},{"alias_kind":"pith_short_16","alias_value":"CKWEZZ2OQXQR52JQ","created_at":"2026-07-07T02:18:36Z"},{"alias_kind":"pith_short_8","alias_value":"CKWEZZ2O","created_at":"2026-07-07T02:18:36Z"}],"graph_snapshots":[{"event_id":"sha256:d227c3d0f11d474d03bb89e161dd3c70ee242a54b6e25231a34f2fd081813f93","target":"graph","created_at":"2026-07-07T02:18:36Z","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/2603.04365/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This paper establishes a comparison theorem for the maximum eigenvalue of a sum of independent random symmetric matrices. The theorem states that the maximum eigenvalue of the matrix sum is dominated by the maximum eigenvalue of a Gaussian random matrix whose statistics match the sum, and it strengthens previous results of this type. Corollaries address the minimum eigenvalue and the spectral norm; the proof strategy also extends to matrix martingale sequences.\n  The comparison methodology is powerful because of the vast arsenal of tools for treating Gaussian random matrices. As applications, ","authors_text":"Joel A. Tropp","cross_cats":["cs.NA","math.NA","math.ST","stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-03-04T18:30:04Z","title":"Comparison theorems for the extreme eigenvalues of a random symmetric matrix"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2603.04365","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":"sha256:24670cb7aaaaa962b89e313f726173d0141183713df6338212f1a639d991c581","target":"record","created_at":"2026-07-07T02:18:36Z","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":"630ea065d2fe48c86f9310fdaa0af49a287f7eb42a2797994f898d8d35f4b377","cross_cats_sorted":["cs.NA","math.NA","math.ST","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-03-04T18:30:04Z","title_canon_sha256":"eee4193c823f266b2c8e677e2b42585b4c42badb8cbdef8d0dd4ac84b4b452ca"},"schema_version":"1.0","source":{"id":"2603.04365","kind":"arxiv","version":3}},"canonical_sha256":"12ac4ce74e85e11ee9309d768a2c5e7ad6e9d4061c2804cc1bd21f5df4178a5a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"12ac4ce74e85e11ee9309d768a2c5e7ad6e9d4061c2804cc1bd21f5df4178a5a","first_computed_at":"2026-07-07T02:18:36.404712Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-07T02:18:36.404712Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TjQ2mjnku+Ku3VzDNuDw6BZ/kw+pH+v7KEIcLLhPgkga7sOuwHRds5WU7EDItiyeuTrJYWNdZDU60QwWm/VbAA==","signature_status":"signed_v1","signed_at":"2026-07-07T02:18:36.405790Z","signed_message":"canonical_sha256_bytes"},"source_id":"2603.04365","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:24670cb7aaaaa962b89e313f726173d0141183713df6338212f1a639d991c581","sha256:d227c3d0f11d474d03bb89e161dd3c70ee242a54b6e25231a34f2fd081813f93"],"state_sha256":"ce2ec6ff7ac72034f807aa834ef67f790d79436b9f5a9db178940ff6e37478b2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nkHJJbAdMdHuT/+PkJOEEcAc2pyPWMOGiT6On/Zm7OeGuWzlEQhVxmoQY1uXoNQMbaqqsHGIc6swPWUi1rO2BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-06T12:12:59.977200Z","bundle_sha256":"c969cfbf120d11033c79b9aec94b0fe25f6847fffe124177ef14a1bdaed3d68a"}}