A Premetric Four-Matrix Route to Emergent Lorentzian Geometry

We investigate a strictly premetric matrix construction in which the fundamental dynamical variables are four noncommuting Hermitian matrices, while no background spacetime metric, vierbein, or coordinate geometry is assumed. Their antisymmetric commutators define a six-dimensional two-form sector endowed only with orientation. We derive algebraic obstructions to several direct metric constructions and formulate candidate geometry through a constitutive response on the two-form sector. Conditional on a closure relation for the induced endomorphism, standard two-form/Urbantke reconstruction yields a conformal Lorentzian metric. We introduce a compact projective finite-N completion whose gauge-fixed representative is a Stiefel manifold and report regulator-free finite-N evidence for strong noncommutative high-P ordering. We further show that a positive Euclidean response, and its Euclidean 1PI inverse, cannot by themselves realize the required Lorentzian closure. The reduction to a massless two-helicity spin-2 sector remains an open dynamical problem.