The fourth neutrino: from a minimal 4 × 3 Dirac extension to the Standard Model

Current data is consistent with both Dirac and Majorana, but the Majorana inverted-ordering branch is being squeezed by cosmological mass-sum bounds, motivating a closer look at the corner where the lightest active mass vanishes. Working within ΛCDM, we study the minimal 4×3 Dirac extension in the heavy-sterile regime |MR| >> |mD|. This matrix forces an exactly massless left-handed eigenstate together with mββ = 0 at all perturbative orders under exact global B−L. The heavy mostly-sterile eigenstate at the free scale |MR| opens heavy neutral lepton (HNL) searches for non-aligned textures. At ms ∼ 1012 GeV, the state lies on the cold-dark-matter contour of ΛCDM under gravitational production at reheating and a νR flavour symmetry, among other inputs not fixed by the Lagrangian. The construction is falsifiable at three independent scales, namely cosmological mass sums along the saturating mlightest = 0 trajectories, 0νββ at the active scale, and a lepton-number-violating (LNV) HNL signature at the heavy scale when non-aligned textures reach observable HNL production. The paper also catalogues the auxiliary keV-window channels (Shi–Fuller resonant production conditional on a primordial lepton asymmetry; Dodelson–Widrow non-resonant production at sub-dominant fractions; Higgs-portal freeze-in as a subdominant correction), the Dirac radiative-decay rate (a factor of one half relative to the Pal–Wolfenstein Majorana formula), and the full set of X-ray, Lyman-α, BBN / CMB ΔNeff, CMB spectral-distortion, direct-detection, and heavy-neutral-lepton constraints across the keV–TeV–superheavy range. This v3.0 is the version transferred to Results in Physics on 2026-09-01 via Elsevier's article transfer service following the Nuclear Physics, Section B post-peer-review decision of 2026-08-28 (Shuang Wu, Scientific Handling Editor). Substantive revisions vs v2.0 (NPB round-2 submitted 2026-06-29) address the R#3 and R#6 peer review comments: abstract and introduction front-load the ΛCDM and heavy-sterile regime scope; Section 2 adds an opposite-regime |MR| << |mD| treatment; Section 6 enumerates the five external inputs the dark-matter reading rests on beyond what the Lagrangian fixes; Discussion adds SHiP and DUNE production and decay processes, a three-way LNV-sensitive HNL discriminator across colliders and beam dumps, UHE ν and γ decay signals at ms ∼ 1012 GeV, and a leptogenesis paragraph evaluating both the standard Majorana route and the Dirac routes against the (4, 3) construction. This combined manuscript supersedes the two per-paper preprints at 10.5281/zenodo.20150279 (structural argument) and 10.5281/zenodo.20150281 (dark-matter phenomenology).