{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:KHA2PVQK37DUGWWPMKICIEKXZM","short_pith_number":"pith:KHA2PVQK","canonical_record":{"source":{"id":"2607.04047","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-07-04T22:28:50Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"d3abf7e45e438d30fce5557b9395b2f34963e644d670a549145ad8d7c487325e","abstract_canon_sha256":"94c3d5d423358bca52ff05d29d02f7cdafdf11311aa4c5c388abb4344191b854"},"schema_version":"1.0"},"canonical_sha256":"51c1a7d60adfc7435acf6290241157cb099a22408cf7552305724a5c5d68cd6d","source":{"kind":"arxiv","id":"2607.04047","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.04047","created_at":"2026-07-07T02:18:54Z"},{"alias_kind":"arxiv_version","alias_value":"2607.04047v1","created_at":"2026-07-07T02:18:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.04047","created_at":"2026-07-07T02:18:54Z"},{"alias_kind":"pith_short_12","alias_value":"KHA2PVQK37DU","created_at":"2026-07-07T02:18:54Z"},{"alias_kind":"pith_short_16","alias_value":"KHA2PVQK37DUGWWP","created_at":"2026-07-07T02:18:54Z"},{"alias_kind":"pith_short_8","alias_value":"KHA2PVQK","created_at":"2026-07-07T02:18:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:KHA2PVQK37DUGWWPMKICIEKXZM","target":"record","payload":{"canonical_record":{"source":{"id":"2607.04047","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-07-04T22:28:50Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"d3abf7e45e438d30fce5557b9395b2f34963e644d670a549145ad8d7c487325e","abstract_canon_sha256":"94c3d5d423358bca52ff05d29d02f7cdafdf11311aa4c5c388abb4344191b854"},"schema_version":"1.0"},"canonical_sha256":"51c1a7d60adfc7435acf6290241157cb099a22408cf7552305724a5c5d68cd6d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-07T02:18:54.903618Z","signature_b64":"rKFfE28ErwJ20NmhtZXSmyYzsRYHvVWdQeo4E+XATAl4fQet5K9dwh8o8sM2uwoNlSKfLmNfPh6jvkm5ZiASBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"51c1a7d60adfc7435acf6290241157cb099a22408cf7552305724a5c5d68cd6d","last_reissued_at":"2026-07-07T02:18:54.902840Z","signature_status":"signed_v1","first_computed_at":"2026-07-07T02:18:54.902840Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.04047","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-07T02:18:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MYx23FgF4EOD+bSQOoCuIsT1CyEN7GbiEs81VgMHSezLztj2/R/aDsDuPYb9EIMtWbsqFUbK7SaQSGAB9WF0AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T20:11:13.623419Z"},"content_sha256":"359779c9ad70e724dbe5e4df6f383fc958486d8d357a7ac6ea972fb21e3ce5bb","schema_version":"1.0","event_id":"sha256:359779c9ad70e724dbe5e4df6f383fc958486d8d357a7ac6ea972fb21e3ce5bb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:KHA2PVQK37DUGWWPMKICIEKXZM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Top Singular Value in Sum-Products of Random Matrices","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Boris Hanin, Kevin Han Huang","submitted_at":"2026-07-04T22:28:50Z","abstract_excerpt":"We study the top singular value for a sum of $m$ independent $n \\times n$ random matrices, each of which is a product of $N$ i.i.d. $n\\times n$ Gaussian matrices. Our main conceptual observation is that when $m,n,N\\rightarrow \\infty$, the top singular value coincides with the partition function in a random energy model at the inverse temperature $\\beta=\\sqrt{2(N-1)/(n\\log m)}$, with energies depending on the ratio $N/n$. We provide several non-asymptotic results making this approximation precise."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.04047","kind":"arxiv","version":1},"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/2607.04047/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":"2026-07-07T02:18:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yRuIU/kvxHoLrEvFf55eZOgUTMv/nkvdir/vlVMcxFzkoZeZVj7HQ0WHDnVB54tiE4wRYWOOTuBTWrvykLURBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T20:11:13.623722Z"},"content_sha256":"f62a0ba69b0dcea52e3880867f44cc823f67922ab3f1ae36ea8f64c1b518b138","schema_version":"1.0","event_id":"sha256:f62a0ba69b0dcea52e3880867f44cc823f67922ab3f1ae36ea8f64c1b518b138"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KHA2PVQK37DUGWWPMKICIEKXZM/bundle.json","state_url":"https://pith.science/pith/KHA2PVQK37DUGWWPMKICIEKXZM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KHA2PVQK37DUGWWPMKICIEKXZM/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":"2026-08-14T20:11:13Z","links":{"resolver":"https://pith.science/pith/KHA2PVQK37DUGWWPMKICIEKXZM","bundle":"https://pith.science/pith/KHA2PVQK37DUGWWPMKICIEKXZM/bundle.json","state":"https://pith.science/pith/KHA2PVQK37DUGWWPMKICIEKXZM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KHA2PVQK37DUGWWPMKICIEKXZM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:KHA2PVQK37DUGWWPMKICIEKXZM","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":"94c3d5d423358bca52ff05d29d02f7cdafdf11311aa4c5c388abb4344191b854","cross_cats_sorted":["math-ph","math.MP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-07-04T22:28:50Z","title_canon_sha256":"d3abf7e45e438d30fce5557b9395b2f34963e644d670a549145ad8d7c487325e"},"schema_version":"1.0","source":{"id":"2607.04047","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.04047","created_at":"2026-07-07T02:18:54Z"},{"alias_kind":"arxiv_version","alias_value":"2607.04047v1","created_at":"2026-07-07T02:18:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.04047","created_at":"2026-07-07T02:18:54Z"},{"alias_kind":"pith_short_12","alias_value":"KHA2PVQK37DU","created_at":"2026-07-07T02:18:54Z"},{"alias_kind":"pith_short_16","alias_value":"KHA2PVQK37DUGWWP","created_at":"2026-07-07T02:18:54Z"},{"alias_kind":"pith_short_8","alias_value":"KHA2PVQK","created_at":"2026-07-07T02:18:54Z"}],"graph_snapshots":[{"event_id":"sha256:f62a0ba69b0dcea52e3880867f44cc823f67922ab3f1ae36ea8f64c1b518b138","target":"graph","created_at":"2026-07-07T02:18:54Z","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/2607.04047/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the top singular value for a sum of $m$ independent $n \\times n$ random matrices, each of which is a product of $N$ i.i.d. $n\\times n$ Gaussian matrices. Our main conceptual observation is that when $m,n,N\\rightarrow \\infty$, the top singular value coincides with the partition function in a random energy model at the inverse temperature $\\beta=\\sqrt{2(N-1)/(n\\log m)}$, with energies depending on the ratio $N/n$. We provide several non-asymptotic results making this approximation precise.","authors_text":"Boris Hanin, Kevin Han Huang","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-07-04T22:28:50Z","title":"Top Singular Value in Sum-Products of Random Matrices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.04047","kind":"arxiv","version":1},"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":"sha256:359779c9ad70e724dbe5e4df6f383fc958486d8d357a7ac6ea972fb21e3ce5bb","target":"record","created_at":"2026-07-07T02:18:54Z","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":"94c3d5d423358bca52ff05d29d02f7cdafdf11311aa4c5c388abb4344191b854","cross_cats_sorted":["math-ph","math.MP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-07-04T22:28:50Z","title_canon_sha256":"d3abf7e45e438d30fce5557b9395b2f34963e644d670a549145ad8d7c487325e"},"schema_version":"1.0","source":{"id":"2607.04047","kind":"arxiv","version":1}},"canonical_sha256":"51c1a7d60adfc7435acf6290241157cb099a22408cf7552305724a5c5d68cd6d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"51c1a7d60adfc7435acf6290241157cb099a22408cf7552305724a5c5d68cd6d","first_computed_at":"2026-07-07T02:18:54.902840Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-07T02:18:54.902840Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rKFfE28ErwJ20NmhtZXSmyYzsRYHvVWdQeo4E+XATAl4fQet5K9dwh8o8sM2uwoNlSKfLmNfPh6jvkm5ZiASBg==","signature_status":"signed_v1","signed_at":"2026-07-07T02:18:54.903618Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.04047","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:359779c9ad70e724dbe5e4df6f383fc958486d8d357a7ac6ea972fb21e3ce5bb","sha256:f62a0ba69b0dcea52e3880867f44cc823f67922ab3f1ae36ea8f64c1b518b138"],"state_sha256":"8bf819e1df9ecbbf824db71672d84dd628dc9f8faf357bceefba43e4c659b44d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"S2qSudU4alorTjNgWXLt2tuZx+GUS9RIntAw8rUacFlLpaL+J7zrQe+nzt8ITDC4Y176ctNcItvH30dVVQ9UCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-14T20:11:13.627033Z","bundle_sha256":"9841ef7e078514c03fd4a5bdb8365ff65cde9d9d90da3cb103da6c1870892da2"}}