{
  "abstract": "μ–τ flavor reflection symmetry (RS) of the PMNS mixing matrix implies |Uμi| = |Uτi|, leading to the atmospheric mixing angle θ23=45° and maximal Dirac CP violation phase, δCP=±90°. Here, we establish an analytical connection between leptonic CP violation and the breaking of μ–τ RS through the effective flavor squared masses mα2=∑i|Uαi|2mi2. We first examine the symmetric condition Δmτμ2≡mτ2−mμ2=0, which constrains θ23 to a highly localized weakly broken-RS geometric band, ±0.123°, centered at θ23=45° and δCP=−90° over the continuous phase interval −180°≤δCP≤0°. The lower edge of the joint T2K–NOvA benchmark interval, θ23(exp)=48.45−2.9°+1.7°, lies above this symmetric boundary (45.55°>45.123°), motivating an extension to finite RS breaking. We therefore introduce the dimensionless RS-breaking indicator (RSBI), β0=Δmτμ2/Δm212≠0, and characterize the departure from RS through the μ–τ asymmetries Δi=|Uμi|2−|Uτi|2 and their flavor-asymmetry quadratic deviation σ2(δCP)=∑iΔi2. This leads us to conjecture that the physical value of the Dirac CP-violating phase corresponds to a minimum of σ2(δCP). At leading-order, the resulting stationary condition provides a closed-form relation between cosδCP and the three mixing angles. This leading order relation lies midway between the partial-symmetry results of Xing and Zhou. Evaluated at the empirical joint T2K–NOvA benchmark, the framework gives |β0| ≃3.8 and δCP=−100−5°+9°, placing the phase in the third quadrant near-maximal CP violation. Including sub-leading corrections associated with the neutrino mass ordering gives the conditional estimates δCP=−101° for Normal Ordering (NO) and δCP=−99° for Inverted Ordering (IO), consistent with the corresponding direct numerical minimizations. These results suggest that δCP may be constrained by collective relations among the neutrino mixing parameters rather than behaving as an entirely independent free parameter.",
  "abstract_shown": true,
  "affiliations": [
    "Department of Physics, University of Milano-Bicocca, Piazza della Scienza 3, 20126 Milano, Italy"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Hector Eduardo Roman",
      "orcid": "0000-0003-3183-9258"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fhewad0d0ren7.bib",
  "cited_by": 0,
  "doi": "10.3390/axioms15110819",
  "fulltext_url": null,
  "id": "fhewad0d0ren7",
  "issns": [
    "2075-1680"
  ],
  "landing_url": "https://www.mdpi.com/2075-1680/15/11/819",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": "MDPI AG",
  "ris": "https://research.draconitic.com/paper/fhewad0d0ren7.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T12:20:59Z",
      "native_id": "10.3390/axioms15110819",
      "source": "crossref",
      "url": "https://www.mdpi.com/2075-1680/15/11/819"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "journal-openalex:F26",
      "subfield": "general"
    }
  ],
  "title": "Near-Maximal CP Violation and μ–τ Flavor Reflection Symmetry Breaking",
  "title_text": "Near-Maximal CP Violation and μ–τ Flavor Reflection Symmetry Breaking",
  "type": "peer-reviewed",
  "updated_at": "2026-11-05T00:20:50Z",
  "url": "https://research.draconitic.com/paper/fhewad0d0ren7",
  "venue": "Axioms",
  "versions": []
}