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For a fixed light-heavy pairing, imposing $m_iU_{\\alpha i}U_{\\beta i}+M_iR_{\\alpha i}R_{\\beta i}=0$ together with $UU^\\dagger+RR^\\dagger=\\mathbf{1}$ yields the general exact solution $U=VD^{-1/2}$ and $R=iVE\\,\\operatorname{diag}\\!\\left(\\sqrt{r_i/(1+r_i)}\\right)$, where $r_i=m_i/M_i$, $D=\\operatorname{diag}(1+r_i)$, $V$ is unitary, and $E=\\operatorname{diag}(\\eta_i)$ with $\\eta_i=\\pm1$. Consequently, $U^\\dagger U$ and $R^\\dagger R$ are diagonal, and disti"},"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":"2607.16658","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"hep-ph","submitted_at":"2026-07-18T06:27:14Z","cross_cats_sorted":[],"title_canon_sha256":"4f8a1892d2f27d2546e1d4c2e87bcc440759a9263acf98228e80a3b7c2bfbde4","abstract_canon_sha256":"f602f7586257576667e9dbefdb6932772525808df26053de2fb511281553eba0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-21T01:20:20.614519Z","signature_b64":"9ArDw3LuCt4Jzj8ke5V7cI0l9XSD/PzvkfqodSQFBuNapryCJKudg2MWw5uSONNNBX4UqmAPUIDUoa7sytRxCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ba2cbef55e01d0fd3095745b804c4825e0d21264ee288e7ac5bba4e55a415554","last_reissued_at":"2026-07-21T01:20:20.613687Z","signature_status":"signed_v1","first_computed_at":"2026-07-21T01:20:20.613687Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Exact constraints on family-separated seesaw relations and their phenomenological consequences","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"hep-ph","authors_text":"Jianlong Lu","submitted_at":"2026-07-18T06:27:14Z","abstract_excerpt":"Working at tree level and at a common renormalization scale, we examine the family-separated seesaw ansatz of Z.-z.~Xing, arXiv:2605.27049v2. For a fixed light-heavy pairing, imposing $m_iU_{\\alpha i}U_{\\beta i}+M_iR_{\\alpha i}R_{\\beta i}=0$ together with $UU^\\dagger+RR^\\dagger=\\mathbf{1}$ yields the general exact solution $U=VD^{-1/2}$ and $R=iVE\\,\\operatorname{diag}\\!\\left(\\sqrt{r_i/(1+r_i)}\\right)$, where $r_i=m_i/M_i$, $D=\\operatorname{diag}(1+r_i)$, $V$ is unitary, and $E=\\operatorname{diag}(\\eta_i)$ with $\\eta_i=\\pm1$. Consequently, $U^\\dagger U$ and $R^\\dagger R$ are diagonal, and disti"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.16658","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/2607.16658/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":"2607.16658","created_at":"2026-07-21T01:20:20.614116+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.16658v1","created_at":"2026-07-21T01:20:20.614116+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.16658","created_at":"2026-07-21T01:20:20.614116+00:00"},{"alias_kind":"pith_short_12","alias_value":"XIWL55K6AHIP","created_at":"2026-07-21T01:20:20.614116+00:00"},{"alias_kind":"pith_short_16","alias_value":"XIWL55K6AHIP2MEV","created_at":"2026-07-21T01:20:20.614116+00:00"},{"alias_kind":"pith_short_8","alias_value":"XIWL55K6","created_at":"2026-07-21T01:20:20.614116+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/XIWL55K6AHIP2MEVORNYATCIEX","json":"https://pith.science/pith/XIWL55K6AHIP2MEVORNYATCIEX.json","graph_json":"https://pith.science/api/pith-number/XIWL55K6AHIP2MEVORNYATCIEX/graph.json","events_json":"https://pith.science/api/pith-number/XIWL55K6AHIP2MEVORNYATCIEX/events.json","paper":"https://pith.science/paper/XIWL55K6"},"agent_actions":{"view_html":"https://pith.science/pith/XIWL55K6AHIP2MEVORNYATCIEX","download_json":"https://pith.science/pith/XIWL55K6AHIP2MEVORNYATCIEX.json","view_paper":"https://pith.science/paper/XIWL55K6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.16658&json=true","fetch_graph":"https://pith.science/api/pith-number/XIWL55K6AHIP2MEVORNYATCIEX/graph.json","fetch_events":"https://pith.science/api/pith-number/XIWL55K6AHIP2MEVORNYATCIEX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XIWL55K6AHIP2MEVORNYATCIEX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XIWL55K6AHIP2MEVORNYATCIEX/action/storage_attestation","attest_author":"https://pith.science/pith/XIWL55K6AHIP2MEVORNYATCIEX/action/author_attestation","sign_citation":"https://pith.science/pith/XIWL55K6AHIP2MEVORNYATCIEX/action/citation_signature","submit_replication":"https://pith.science/pith/XIWL55K6AHIP2MEVORNYATCIEX/action/replication_record"}},"created_at":"2026-07-21T01:20:20.614116+00:00","updated_at":"2026-07-21T01:20:20.614116+00:00"}