A minimal Dirac extension of the Standard Model neutrino sector: rank-three structure forces a vanishing lightest active mass

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. 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, well below current and planned cosmological sensitivity. The cosmological neutrino mass sum is therefore tightly determined: 0.059 eV in normal ordering, 0.10 eV in inverted ordering, with the proposal preferring NO under conservative cosmological priors. The model predicts mββ = 0 in the strict EFT and mββ ≲ 10−5 eV including the same Planck-suppressed corrections; either way, 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. A null next-generation result at the most aggressive 9-meV sensitivity is consistent with the proposal, excludes Majorana inverted ordering, and leaves the Majorana normal-ordering surviving region with small mlightest and phase cancellation intact. The asymmetric structure of the 0νββ test is the sharpest empirical test of this class of model at next-generation sensitivities. The paper works the explicit singular value decomposition of the 4×3 mass matrix, computes the spectrum at heavy MR via Schur complement, exhibits a Yukawa texture reproducing measured PMNS angles, and verifies radiative stability of the rank-three structure under exact B−L protection in the EFT. The proposal differs structurally from both the Type-I seesaw / νMSM (which breaks B−L and predicts non-zero mββ) and from the pure three-νR Dirac extension (which produces no rank-three forced zero). The present paper establishes the structural setup. The phenomenology of the heavy mostly-sterile sector at scale |MR|, including the Conditional CDM Proposition for the superheavy gravitational-production branch, the auxiliary keV-window channels, and the X-ray / Lyman-α / BBN / CMB / direct-detection / heavy-neutral-lepton constraint map across the keV–superheavy mass range, is developed in the companion paper (Zenodo DOI: 10.5281/zenodo.20150281).