{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:K4IKOPTBTAH7YNJE2Q5K32L55Z","short_pith_number":"pith:K4IKOPTB","schema_version":"1.0","canonical_sha256":"5710a73e61980ffc3524d43aade97dee5dba798f78d48d60d5b9d50d11e573e9","source":{"kind":"arxiv","id":"2607.07617","version":1},"attestation_state":"computed","paper":{"title":"Loop Equations Characterize Random Matrix Statistics","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Jiaoyang Huang, Paul Bourgade","submitted_at":"2026-07-08T16:33:33Z","abstract_excerpt":"We prove that the universal local point processes of random matrix theory are characterized by their loop equation hierarchies. More precisely, for every rational $\\beta>0$, the $\\mathrm{Sine}_{\\beta}$ point process is the unique solution of the bulk loop equation hierarchy, and the $\\mathrm{Airy}_{\\beta}$ point process is the unique solution of the edge loop equation hierarchy.\n  These uniqueness results provide a direct route to universality: it suffices to verify the corresponding approximate loop equations for the ensemble. In many models, these equations follow from local laws and integra"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2607.07617","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-07-08T16:33:33Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"6ab0c3a22703669fc8c3e23f395b1f08f1f2930f18cd0740ab41a5c241f7073b","abstract_canon_sha256":"dd3e831b0d83a4deac5f02a599544888abdab1a4006566707013fde2aea9f030"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-09T01:20:36.892607Z","signature_b64":"C31Au77RR2x6w6/Zv2Id32aY88zKdXQzlTNddpN1bHp881c/qRn7p4SCJ4JDf/TqQZcmK8Gja0g0q5E5oFp2CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5710a73e61980ffc3524d43aade97dee5dba798f78d48d60d5b9d50d11e573e9","last_reissued_at":"2026-07-09T01:20:36.892048Z","signature_status":"signed_v1","first_computed_at":"2026-07-09T01:20:36.892048Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Loop Equations Characterize Random Matrix Statistics","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Jiaoyang Huang, Paul Bourgade","submitted_at":"2026-07-08T16:33:33Z","abstract_excerpt":"We prove that the universal local point processes of random matrix theory are characterized by their loop equation hierarchies. More precisely, for every rational $\\beta>0$, the $\\mathrm{Sine}_{\\beta}$ point process is the unique solution of the bulk loop equation hierarchy, and the $\\mathrm{Airy}_{\\beta}$ point process is the unique solution of the edge loop equation hierarchy.\n  These uniqueness results provide a direct route to universality: it suffices to verify the corresponding approximate loop equations for the ensemble. In many models, these equations follow from local laws and integra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.07617","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/2607.07617/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2607.07617","created_at":"2026-07-09T01:20:36.892116+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.07617v1","created_at":"2026-07-09T01:20:36.892116+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.07617","created_at":"2026-07-09T01:20:36.892116+00:00"},{"alias_kind":"pith_short_12","alias_value":"K4IKOPTBTAH7","created_at":"2026-07-09T01:20:36.892116+00:00"},{"alias_kind":"pith_short_16","alias_value":"K4IKOPTBTAH7YNJE","created_at":"2026-07-09T01:20:36.892116+00:00"},{"alias_kind":"pith_short_8","alias_value":"K4IKOPTB","created_at":"2026-07-09T01:20:36.892116+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/K4IKOPTBTAH7YNJE2Q5K32L55Z","json":"https://pith.science/pith/K4IKOPTBTAH7YNJE2Q5K32L55Z.json","graph_json":"https://pith.science/api/pith-number/K4IKOPTBTAH7YNJE2Q5K32L55Z/graph.json","events_json":"https://pith.science/api/pith-number/K4IKOPTBTAH7YNJE2Q5K32L55Z/events.json","paper":"https://pith.science/paper/K4IKOPTB"},"agent_actions":{"view_html":"https://pith.science/pith/K4IKOPTBTAH7YNJE2Q5K32L55Z","download_json":"https://pith.science/pith/K4IKOPTBTAH7YNJE2Q5K32L55Z.json","view_paper":"https://pith.science/paper/K4IKOPTB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.07617&json=true","fetch_graph":"https://pith.science/api/pith-number/K4IKOPTBTAH7YNJE2Q5K32L55Z/graph.json","fetch_events":"https://pith.science/api/pith-number/K4IKOPTBTAH7YNJE2Q5K32L55Z/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/K4IKOPTBTAH7YNJE2Q5K32L55Z/action/timestamp_anchor","attest_storage":"https://pith.science/pith/K4IKOPTBTAH7YNJE2Q5K32L55Z/action/storage_attestation","attest_author":"https://pith.science/pith/K4IKOPTBTAH7YNJE2Q5K32L55Z/action/author_attestation","sign_citation":"https://pith.science/pith/K4IKOPTBTAH7YNJE2Q5K32L55Z/action/citation_signature","submit_replication":"https://pith.science/pith/K4IKOPTBTAH7YNJE2Q5K32L55Z/action/replication_record"}},"created_at":"2026-07-09T01:20:36.892116+00:00","updated_at":"2026-07-09T01:20:36.892116+00:00"}