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

We study the minimal Dirac extension of the Standard Model neutrino sector defined by adding a single gauge-singlet left-handed field NL paired with the three right-handed neutrinos νR1,2,3 by the Dirac mass operator MRa N̅L νRa alongside the active Yukawa coupling. Every term preserves B−L as a global symmetry, anomaly-free against the Standard Model gauge currents. 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. The minimal 4×3 Dirac extension has a mass matrix of rank at most three, so the kernel of M† is at least one-dimensional and the corresponding left-handed combination is exactly massless at all perturbative orders in the effective field theory, with mββ = 0 likewise an EFT-level identity under exact global B−L. The construction also introduces a heavy mostly-sterile mass eigenstate at the non-zero singular value σ1 ≈ |MR| with |MR| a free scale parameter. This opens heavy neutral lepton searches for textures with mD · M̂R ≠ 0. The same heavy state at ms ∼ 1012 GeV admits a cold dark matter reading contingent on gravitational production at reheating, a νR flavour symmetry, and further inputs not fixed by the Lagrangian; an explicit benchmark on the spin-resolved PIDM closure curve places the abundance at ΩDMh2 = 0.12 with the heavy state effectively collisionless across cosmological history. Since no parameter in the construction can evade mlightest = 0 or mββ = 0, the rank-three structure is testable at three independent scales: cosmological mass sums along the saturating mlightest = 0 trajectories; 0νββ at the active scale (any positive signal at mββ ≳ 21 meV excluding the extension across all nuclear-matrix-element systematics); and a background-subtracted same-sign dilepton signal at the heavy scale for non-aligned textures with 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 v2.0 is the round-2 revision submitted to Nuclear Physics, Section B (Ms. NPB-D-26-01049R1, 2026-06-29) following the post-peer-review decision of 2026-06-22 (Shuang Wu, Scientific Handling Editor). Currently under editorial-office screening. This combined manuscript supersedes the two per-paper preprints at 10.5281/zenodo.20150279 (structural argument) and 10.5281/zenodo.20150281 (dark-matter phenomenology), integrating both with citation corrections, structural updates per editor and reviewer feedback, and tone refinements.