From Finite Color Spectral Data to Spacetime QCD: Part II—Almost-Commutative Color Geometry and Yang–Mills–Dirac Dynamics

This paper is archived as a speculative research work.Paper I constructed a finite real-even noncommutative-geometric representation of the independently established Entanglement-Algebraic Spacetime Scalar-Field color sector. Its endpoint is the finite package F_ col^ EAS=( M_3( ), ^12, ,D_F,J_F, _F), with represented color group SU(3), quark and antiquark carriers 3 and 3, one color-singlet mass modulus in the minimal generic-quark factor, and vanishing finite one-form space. The present paper begins the spacetime construction from those data without assuming the Yang–Mills connection that it must reproduce. On a qualified four-dimensional Riemannian Spin spectral host, the almost-commutative product has algebra C^∈fty(M_E) ( M_3( )) and Dirac operator D=D_E I+ _E D_F. Because [D_F, (A_F^ col)]=0, every nontrivial product one-form originates in the spacetime differential sector; no finite color scalar fluctuation is generated. The direct and opposite finite actions combine pointwise to a single matrix-valued connection B_ =M_ -y_ I_3∈ u(3) on the quark triplet and the conjugate connection -B_ ^T on the antiquark carrier. Thus the raw almost-commutative gauge algebra is u(3), in exact agreement with the raw represented unitary quotient found in Paper I. The independently established determinant-one SF color restriction selects the corresponding connection-level color sector by the canonical traceless projection B_ col=B- 13 (B)I_3∈ ^1(M_E, su(3)); under SU(3) transformations this projected one-form transforms as an SU(3) connection. Thus the central u(1) component of the raw product connection is not claimed to vanish merely from restricting the gauge transformations; it is excluded by the established SF-compatible color-interface selection. This construction fixes the provenance of the spacetime color connection and preserves the Paper-I separation between spectral output and imported color compatibility. From the resulting su(3) connection we derive the non-Abelian curvature F_B=dB+B B, its adjoint gauge transformation law, the conjugate antiquark curvature and the covariant Bianchi identity. The same connection enters the fluctuated Euclidean Dirac operator on both quark chiralities, yielding one vectorlike color-triplet Dirac coupling with the Paper-I mass modulus and no finite color scalar. The bosonic spectral-action calculation fixes the finite color trace factor and the normalization relation that puts the gluon kinetic term into canonical Euclidean Yang–Mills form without predicting a numerical strong coupling. Combining the bosonic and fermionic sectors gives the Euclidean QCD action for one generic color triplet; variation yields the sourced Yang–Mills equation, the Euclidean quark equation and a covariantly conserved color current. A standard linear covariant gauge then gives the field-dependent non-Abelian Faddeev–Popov operator, its interacting adjoint ghost action and the conventional Euclidean QCD functional-integral entry. Finally, a separately qualified four-dimensional Lorentzian Spin/Krein branch carries the same finite SU(3) representation and common g_s normalization into the classical Lorentzian Yang–Mills–Dirac action and its sourced field equations. No global Wick-rotation or Riemannian–Lorentzian spectral-equivalence theorem is assumed.