We construct a quantum field theory with gauge group SU(3) on the Kuhn triangulation of Z^4 and prove that it has a strictly positive mass gap Delta >= 0.025 > 0. The construction satisfies a set of pentachoric axioms (PA0-PA5), which compare to the Osterwalder-Schrader (OS) axioms as follows: strictly stronger in four of seven directions (compactness, six-fold reflection positivity, constructive clustering, Gibbs uniqueness), equal in one (permutation symmetry), and formally weaker in two (discrete translations Z^4 vs R^4; point-group rotations vs exact SO(4)). Both weaker directions are mitigated: the Givens stability theorem bounds the residual at = 0, and a unique vacuum, without invoking SO(4) invariance. The theory rests on one observation: the density at each vertex of the pentachoric lattice (K5, the complete graph on 5 vertices, the unique 4-simplex in d=4) is a 3x3 Hermitian matrix rho_v with eigenvalues in [0,1]. Its nine degrees of freedom decompose as 1 trace (U(1)) + 8 traceless (su(3) gluon field). The gauge connection on each edge is the spectral comparator U_ij = P_i^dagger P_j in SU(3), derived from the eigenbases of adjacent density matrices. The cubic vertex Tr(delta rho^3) produces the exact structure constants f^abc of SU(3). The effective action, obtained by integrating out eigenvalue fluctuations, has the Wilson plaquette form Delta S = beta Tr(I - W_f) with beta = 0.145 (derived, not input), giving confinement with string tension sigma = 3.73. Reflection positivity (OS2) is proved on the full 4D Kuhn lattice via transposition reflections theta: x_mu x_nu in S4, giving six independent hyperplanes; Cartesian reflections fail (14/30 edge vectors broken). Dobrushin uniqueness (gamma = 0.176 [T2]. The gap has a topological origin: the first Betti number beta_1(Delta^4) = 0 (the 4-simplex is contractible), so the Hodge Laplacian on 1-forms has ker(Delta_1) = {0} — every gauge mode is massive, with no harmonic 1-form to protect a zero mode. The Givens stability theorem bounds approximate Lorentz covariance: every R in SO(4) decomposes into <= 6 = C(4,2) Givens rotations, each belonging to a hyperplane stabiliser; a spectral non-amplification argument gives |W_n(f) - W_n(f o R)| <= 6 C_n (l_P/L)^4 gamma^{L/l_P}, with all constants explicit [T1]. Zero free parameters (alpha* = 1/(4 ln 2) from Bekenstein-Hawking). Companion script: 393 tests, 30 blocks, all PASS.