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Over the subfamily $d\\le X$, their joint distribution converges in total variation to the product of two copies of the Cohen--Lenstra--Gerth distribution, with error bounded by a negative power of $\\log\\log X$. We further conjecture that the corrected $2$-primary groups $2\\operatorname{Cl}_{\\mathbb{Q}(\\sqrt{-d})}[2^"},"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":"2608.00387","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-08-01T01:55:26Z","cross_cats_sorted":[],"title_canon_sha256":"6c0f68e9d88006af6fe8406447f8ef5b00e035ae44832a1440fe634d79f9962a","abstract_canon_sha256":"489c7b9e09037ee67fef2a64aa0e071ec0b7f295512506b678fcf0b0a62c8d09"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-04T00:35:37.626788Z","signature_b64":"s62xXHNwa0Jeoeh60L+4mdVPTYT/XH+uMU8e7AbZL2IeCf2Zf1EHGhF4OQJFP3p2xgKCduMuTIsS3sdiwzYDCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"072000abb5ee69f495bc25da3fa18223fc83e3954262ceaed51a76d3e3707837","last_reissued_at":"2026-08-04T00:35:37.625023Z","signature_status":"signed_v1","first_computed_at":"2026-08-04T00:35:37.625023Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic independence of class-group 4-ranks in correlated pairs of imaginary quadratic fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Xiuwu Zhu, Yue Xu","submitted_at":"2026-08-01T01:55:26Z","abstract_excerpt":"Fix a squarefree integer $d_0>1$, and let $d$ range over the positive squarefree integers coprime to $2d_0$. Although $\\mathbb{Q}(\\sqrt{-d})$ and $\\mathbb{Q}(\\sqrt{-d_0d})$ share all variable ramified primes, we prove that their class-group $4$-ranks are asymptotically independent. Over the subfamily $d\\le X$, their joint distribution converges in total variation to the product of two copies of the Cohen--Lenstra--Gerth distribution, with error bounded by a negative power of $\\log\\log X$. We further conjecture that the corrected $2$-primary groups $2\\operatorname{Cl}_{\\mathbb{Q}(\\sqrt{-d})}[2^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.00387","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/2608.00387/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":"2608.00387","created_at":"2026-08-04T00:35:37.626296+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.00387v1","created_at":"2026-08-04T00:35:37.626296+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.00387","created_at":"2026-08-04T00:35:37.626296+00:00"},{"alias_kind":"pith_short_12","alias_value":"A4QABK5V5ZU7","created_at":"2026-08-04T00:35:37.626296+00:00"},{"alias_kind":"pith_short_16","alias_value":"A4QABK5V5ZU7JFN4","created_at":"2026-08-04T00:35:37.626296+00:00"},{"alias_kind":"pith_short_8","alias_value":"A4QABK5V","created_at":"2026-08-04T00:35:37.626296+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/A4QABK5V5ZU7JFN4EXND7IMCEP","json":"https://pith.science/pith/A4QABK5V5ZU7JFN4EXND7IMCEP.json","graph_json":"https://pith.science/api/pith-number/A4QABK5V5ZU7JFN4EXND7IMCEP/graph.json","events_json":"https://pith.science/api/pith-number/A4QABK5V5ZU7JFN4EXND7IMCEP/events.json","paper":"https://pith.science/paper/A4QABK5V"},"agent_actions":{"view_html":"https://pith.science/pith/A4QABK5V5ZU7JFN4EXND7IMCEP","download_json":"https://pith.science/pith/A4QABK5V5ZU7JFN4EXND7IMCEP.json","view_paper":"https://pith.science/paper/A4QABK5V","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.00387&json=true","fetch_graph":"https://pith.science/api/pith-number/A4QABK5V5ZU7JFN4EXND7IMCEP/graph.json","fetch_events":"https://pith.science/api/pith-number/A4QABK5V5ZU7JFN4EXND7IMCEP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/A4QABK5V5ZU7JFN4EXND7IMCEP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/A4QABK5V5ZU7JFN4EXND7IMCEP/action/storage_attestation","attest_author":"https://pith.science/pith/A4QABK5V5ZU7JFN4EXND7IMCEP/action/author_attestation","sign_citation":"https://pith.science/pith/A4QABK5V5ZU7JFN4EXND7IMCEP/action/citation_signature","submit_replication":"https://pith.science/pith/A4QABK5V5ZU7JFN4EXND7IMCEP/action/replication_record"}},"created_at":"2026-08-04T00:35:37.626296+00:00","updated_at":"2026-08-04T00:35:37.626296+00:00"}