Dark-matter phenomenology of the heavy mostly-sterile state in the minimal 4x3 Dirac extension of the Standard Model

The companion paper (Zenodo DOI: 10.5281/zenodo.20150279) introduces a 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 paired by the active Yukawa, with the singlet Dirac mass operator MRa N̅L νRa. The 4×3 rank-three structure of the neutral-fermion mass matrix forces a vanishing lightest active mass and mββ = 0 exactly, while structurally supplying a heavy mostly-sterile eigenstate at mass |MR| as a candidate cosmological dark-matter particle. The renormalisable Dirac extension supplies the heavy mostly-sterile candidate without Peccei–Quinn symmetry, stabilising parity, Majorana mass operators, gauge portals, scotogenic loops, extra dimensions, or primordial-black-hole collapse assumptions; the CDM identification then follows conditionally after imposing the gravitational-production branch and a lifetime-compatible heavy-column texture. We establish a Conditional CDM Proposition for this candidate. Combining the rank-three argument of the companion paper, the Planckian gravitational-production (PIDM) yield equation at reheating, the gamma-ray decay-line lifetime bound on the active-sterile mixing, and standard kinematic redshifting yields four properties along the spin-resolved PIDM closure curve at the canonical superheavy benchmark ms = 1012 GeV with TRH ∼ 2×109–4×1010 GeV across the conventional reheating-phase band: the heavy state exists at |MR| as the non-zero singular value of the 4×3 rank-three mass matrix; its abundance saturates ΩDMh2 = 0.120 at full DM along the closure curve; the dominant inelastic mass-changing scattering channels are deep-inelastic at √s ≈ ms with combined propagator-level cross section σsN, inelprop ∼ 4×10−111 cm2 under the gamma-ray lifetime bound, leaving Γscat/H << 1 qualitatively; and the candidate is cold with negligible free-streaming length at structure formation. The proposition is conditional on residues that the companion structural paper does not derive: cosmological branch selection, heavy-column texture-selection / fine-tuning (mDa 3 ≲ 1.9×10−20 eV at |MR| = 1012 GeV, fifteen orders below the Planck-suppressed B−L-violation floor; confirmed paper-local at three independent substrate-side levels), the heavy mass scale |MR|, the reheating temperature TRH, the spin-resolved PIDM normalization and produced spectrum, the minimality of the singlet sector, and the high-energy propagator-level cosmological-scattering closure. 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%, and Higgs-portal freeze-in as a subdominant correction), the Dirac radiative-decay rate (a factor of ½ relative to the Majorana Pal–Wolfenstein 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. The Dirac character of the construction carries a positive same-sign dilepton signal at heavy-neutral-lepton discovery as an asymmetric falsifier at the heavy mass-eigenstate level, complementing the asymmetric 0νββ falsifier of the companion paper at the active-neutrino level; the pair of papers offers two independent asymmetric falsification structures of the same Dirac-vs-Majorana question at two mass scales separated by 9–14 orders of magnitude.