PRIM: From Prime Arithmetic to Matter and Lorentz Geometry

PRIM asks how much of the architecture normally introduced separately as matter, spacetime geometry and gravity can be generated from one pre-geometric arithmetic response system. The starting carrier is the exact prime-valuation representation of positive rationals together with the zeta-organized positive counting cone. A declared coefficient-faithful, orientation-sensitive two-step response closes rigidly on five complex directions, E5 = E3 ⊕ E2, with dimensions 3+2. The two descendants are kept typed and are not identified. The internal completion yields the even algebra M3(C) ⊕ M2(C), a determinant-constrained Abelian charge ray and a chiral sixteen-state exterior carrier. A separately regenerated two-dimensional response is completed by all self-adjoint observables, giving Herm2 with Lorentz-signature determinant form. The same arithmetic counting successor supplies a typed Parent/carry connector. On a regular nondegenerate branch the operator-geometry sector admits a Lorentz-Cartan realization; under the stated scaling hypotheses the curvature-linear response reaches Palatini-Holst / Einstein-Cartan, with Einstein gravity on the spinless infrared branch. The manuscript also records a bounded spherical branch with a finite anti-trapped window and a laboratory-scale Parent-anisotropy prediction candidate. Global Actual-State selection, universal regularity, absolute scale fixing, complete Yang-Mills dynamics and nonlocal UV completion remain open.