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Under the matching condition \\[ \\frac{u}{w^2}=\\frac{u'}{(w')^2}, \\] we prove that the resulting hive law is close, in relative entropy, to a GUE hive law. More precisely, if \\[ a^2=w^2+u^2,\\qquad b^2=(w')^2+(u')^2, \\] then the produced hive density $q_n$ satisfies \\[ D_{\\mathrm{"},"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":"2607.04138","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-07-05T06:43:45Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"a94955b1fe9cd27621406b8116d630f5a0cc8d44285d639ac71e94073f26d52c","abstract_canon_sha256":"adec9f12636130e2c2784f1270d990f7587f5be362389ae4a6b590b4da495066"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-07T02:18:59.192028Z","signature_b64":"ocrt54TAsw2rQTmhXc/Bg3N7v6odBQMXnUxaqr7/32s3Wcu61L1FqvBalKcV6IA5Lvqk1+eq+4MoY+ATiecSDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a76de043218d885f0a1cd95664701300aa6630e7de18cb678d68660bf5241813","last_reissued_at":"2026-07-07T02:18:59.191204Z","signature_status":"signed_v1","first_computed_at":"2026-07-07T02:18:59.191204Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hives from deformed GUE minor processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Hariharan Narayanan","submitted_at":"2026-07-05T06:43:45Z","abstract_excerpt":"We construct random hives from deformed GUE minor processes. Starting from two independent diagonally deformed GUE matrices \\[ X=\\sqrt{n}(wG+uD),\\qquad Y=\\sqrt{n}(w'G'+u'D'), \\] where \\(D,D'\\) are diagonal and have GUE spectra, we use their minor processes to form a double hive and then apply the octahedron recurrence. Under the matching condition \\[ \\frac{u}{w^2}=\\frac{u'}{(w')^2}, \\] we prove that the resulting hive law is close, in relative entropy, to a GUE hive law. More precisely, if \\[ a^2=w^2+u^2,\\qquad b^2=(w')^2+(u')^2, \\] then the produced hive density $q_n$ satisfies \\[ D_{\\mathrm{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.04138","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/2607.04138/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":"2607.04138","created_at":"2026-07-07T02:18:59.191334+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.04138v1","created_at":"2026-07-07T02:18:59.191334+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.04138","created_at":"2026-07-07T02:18:59.191334+00:00"},{"alias_kind":"pith_short_12","alias_value":"U5W6AQZBRWEF","created_at":"2026-07-07T02:18:59.191334+00:00"},{"alias_kind":"pith_short_16","alias_value":"U5W6AQZBRWEF6CQ4","created_at":"2026-07-07T02:18:59.191334+00:00"},{"alias_kind":"pith_short_8","alias_value":"U5W6AQZB","created_at":"2026-07-07T02:18:59.191334+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/U5W6AQZBRWEF6CQ43FLGI4ATAC","json":"https://pith.science/pith/U5W6AQZBRWEF6CQ43FLGI4ATAC.json","graph_json":"https://pith.science/api/pith-number/U5W6AQZBRWEF6CQ43FLGI4ATAC/graph.json","events_json":"https://pith.science/api/pith-number/U5W6AQZBRWEF6CQ43FLGI4ATAC/events.json","paper":"https://pith.science/paper/U5W6AQZB"},"agent_actions":{"view_html":"https://pith.science/pith/U5W6AQZBRWEF6CQ43FLGI4ATAC","download_json":"https://pith.science/pith/U5W6AQZBRWEF6CQ43FLGI4ATAC.json","view_paper":"https://pith.science/paper/U5W6AQZB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.04138&json=true","fetch_graph":"https://pith.science/api/pith-number/U5W6AQZBRWEF6CQ43FLGI4ATAC/graph.json","fetch_events":"https://pith.science/api/pith-number/U5W6AQZBRWEF6CQ43FLGI4ATAC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/U5W6AQZBRWEF6CQ43FLGI4ATAC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/U5W6AQZBRWEF6CQ43FLGI4ATAC/action/storage_attestation","attest_author":"https://pith.science/pith/U5W6AQZBRWEF6CQ43FLGI4ATAC/action/author_attestation","sign_citation":"https://pith.science/pith/U5W6AQZBRWEF6CQ43FLGI4ATAC/action/citation_signature","submit_replication":"https://pith.science/pith/U5W6AQZBRWEF6CQ43FLGI4ATAC/action/replication_record"}},"created_at":"2026-07-07T02:18:59.191334+00:00","updated_at":"2026-07-07T02:18:59.191334+00:00"}