From Scalar-Field Color to a Finite Noncommutative Geometry of QCD: Part I—Finite Color Spectral Data

This paper is archived as a speculative research work.We construct a finite real-even noncommutative-geometric color package from the independently certified Entanglement-Algebraic Spacetime Scalar-Field color sector. The upstream results already provide the carrier C^3, the algebra M_3( C), su(3) and SU(3), and the fundamental and conjugate color representations. The question here is whether those data admit a finite real-even spectral representation without importing additional physical structure into the Scalar Fields. On the selected two-grading-copy fundamental carrier, M_3( C) alone cannot support a nontrivial commuting opposite M_3( C) action. Within the finite-dimensional direct-sum C^*-algebra comparison class considered here, the minimal completion is therefore A_F^ col= C M_3( C). On the resulting 12-dimensional carrier we construct a faithful representation, KO-dimension-six real structure and grading, and classify every finite Dirac operator on this fixed package satisfying self-adjointness, oddness, reality and first order. The spectral axioms leave an arbitrary A∈ M_3( C) in the quark chiral block; compatibility with the previously established irreducible vectorlike SU(3) action then forces A=dI_3. The color-preserving family has one nonnegative mass modulus and vanishing finite one-form space, so it generates no finite color scalar fluctuation. The raw represented real-spectral unitary group is U(3), not SU(3). Imposing determinant one on the represented fundamental color action, as required by the established SF color group, yields exactly SU(3) acting as 3 on quark slots and 3 on their real-conjugate partners. The resulting finite package supplies the real-even internal data needed for the almost-commutative spacetime construction of Paper II.