A minimal 4x3 Dirac extension of the Standard Model: rank-three structure, vanishing lightest active mass, and a Conditional CDM Proposition for the heavy mostly-sterile state

We consider the minimal Dirac extension of the Standard Model neutrino sector defined by adding a single gauge-singlet left-handed field NL to the three right-handed neutrinos νR1,2,3, paired by the Dirac mass operator MRa N̅L νRa in addition to the active Yukawa coupling. Every term preserves B−L as a global symmetry, anomaly-free against the Standard Model gauge currents. Part I (structural setup). The resulting 4×3 complex neutral-fermion mass matrix has 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 orders in the effective field theory. Including only the Planck-suppressed quantum-gravity violation of global symmetries permitted by the standard "no global symmetries in quantum gravity" expectation, the lightest active mass is bounded by mlightest ≲ v2/MPl ≈ 10−5 eV. The model predicts mββ = 0 in the strict EFT and mββ ≲ 10−5 eV including the same Planck-suppressed corrections; any positive next-generation neutrinoless double-beta-decay signal at sensitivity mββ ≳ 21 meV (NME-robust at LEGEND-1000 / nEXO) excludes the extension under all NME systematics. Part II (dark-matter phenomenology). The same rank-three structure supplies a heavy mostly-sterile mass eigenstate at the non-zero singular value σ1 ≈ |MR|. We establish a Conditional CDM Proposition for this heavy state. Combining the rank-three argument, the spin-resolved Planckian-Interacting-Massive-Particle gravitational-production yield equation at reheating, the gamma-ray decay-line cosmological-lifetime bound on the active-sterile mixing, and standard kinematic redshifting yields four ΛCDM properties along the spin-resolved PIDM closure curve at the canonical superheavy benchmark ms = 1012 GeV with TRH ∼ 2×109–4×1010 GeV: the heavy state exists at |MR|; its abundance saturates ΩDMh2 = 0.120 at full DM along the closure curve; the dominant inelastic mass-changing scattering channels carry combined propagator-level cross section σsN, inelprop ∼ 4×10−126 cm2 with Γscat/H << 1 across cosmological history; and the candidate is cold with negligible free-streaming length at structure formation. The proposition is conditional on seven residues catalogued explicitly in §16.4. Two-scale falsifier. The Dirac character of the construction carries an asymmetric falsifier at the heavy-state scale: a positive same-sign dilepton signal at any future heavy-neutral-lepton discovery falsifies the Dirac interpretation. Combined with the asymmetric 0νββ falsifier at the active-neutrino scale, the construction offers two independent asymmetric falsification structures of the same Dirac-vs-Majorana question at two mass scales separated by 7–13 orders of magnitude across the heavy-neutral-lepton mass window (MeV to TeV). The paper also documents the auxiliary keV-window channels (Shi–Fuller resonant production conditional on a primordial lepton asymmetry; Dodelson–Widrow non-resonant production as a sub-dominant fractional contribution at fs ∈ [4%, 13%]; Higgs-portal freeze-in as a subdominant correction), the Dirac radiative-decay rate (a factor of ½ 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 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 and tone refinements. Currently under review at Annals of Physics (manuscript 83593, submitted 2026-05-21).