Spectral gap of a compact density-matrix transfer operator on the Kuhn triangulation of Z⁴

We study the elementary cell of the Kuhn (Freudenthal) triangulation of Z⁴: five 3×3 density matrices on the vertices of K₅, coupled by a Jaynes entropy and a Dirichlet gradient on the Pauli domain 0 ≤ ρ ≤ I₃. We prove that the causal transfer kernel of this cell has a strictly positive spectral gap, Δ > 0, for every coupling α* > 0 (T1 = proved, T2 = numerically verified). No expansion and no small parameter enter: the gap follows, by classical positive-kernel methods, from three verified structural properties of the model — a continuous, strictly positive kernel on a compact domain, symmetric under causal reflection. A topological complement (β₁(Δ⁴) = 0) excludes gapless winding modes. At α* = 1/(4 ln 2), Gauss–Legendre diagonalisation gives Δ = 0.93689 per cell (1.00463 under the canonical splitting) (T2); the scalar sector reduces exactly to a one-dimensional integral operator whose gap is the parameter-free spectral constant δ* = 0.93408217792 of K₅ (T1). A Birkhoff bound yields the floor Δ ≥ −ln tanh(Smax/2). Every numerical claim is scripted and reproducible (100 tests, all passing).