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We show that these intriguing possibilities can be a very natural consequence of the {\\it translational} $\\mu$-$\\tau$ reflection symmetry -- the effective Majorana neutrino mass term keeps invariant under the transformations $\\nu^{}_{e \\rm L} \\to (\\nu^{}_{e \\rm L})^c + U^*_{e i} z^{c}_\\nu$, $\\nu^{}_{\\mu \\rm L} \\to (\\nu^{}_{\\tau \\rm L})^c + U^*_{\\tau "},"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":"2207.08568","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"hep-ph","submitted_at":"2022-07-18T12:52:35Z","cross_cats_sorted":[],"title_canon_sha256":"fd9b62c3bae50067d4428fdee4c72da99f13af27e5eaa919092978670e9104cd","abstract_canon_sha256":"cb3022b57e394c06139df2ae285f0ca53758cb485cd7e320e36566aa46867545"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:52:57.305456Z","signature_b64":"6DCnWG/YUDw4CBzuWvgNNGDNfuamL9AfM7ra0lLlhV6gJgn8ysQjh4I7npTEwfMPMjemHqtXXax6nCWjdUhJAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"48cf65f6a3699608c4935ecca3c0962020480835b362477a7d7f281341052d6f","last_reissued_at":"2026-07-05T05:52:57.304975Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:52:57.304975Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The translational $\\mu$-$\\tau$ reflection symmetry of Majorana neutrinos","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"hep-ph","authors_text":"Zhi-zhong Xing","submitted_at":"2022-07-18T12:52:35Z","abstract_excerpt":"The present neutrino oscillation data allow $m^{}_1 = 0$ (or $m^{}_3 = 0$) for the neutrino mass spectrum and support $\\theta^{}_{23} \\simeq \\pi/4$ and $\\delta \\simeq -\\pi/2$ as two good approximations for the PMNS lepton flavor mixing matrix $U$. 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