Project T Volume LI : The Crystal Universality Theorem Yang–Mills Mass Gap for All Compact Simple Gauge Groups

We extend the Crystal Construction of Volume L from SU(3) to all compact simple Lie groups. The key observation is that Kashiwara’s crystal basis theory provides positivity (P1), canonicity (P2), and tensor compatibility (P3) for every symmetrizable Kac–Moody algebra — not just type Ã₂. Since every finite Weyl group W(g) embeds in a symmetric group Sₙ, the entire constructive machinery (mass ladder, spectral convergence, uniform Schwinger bounds, Osterwalder–Schrader axioms) transfers to arbitrary type. We prove: for any compact simple gauge group G, the crystal construction produces a Wightman QFT on ℝ⁴ with mass gap Δ(G) = π² + μ²(G) > 0, where μ²(G) = λ₁(W(g)) is the spectral gap of the Weyl group Cayley graph. The separation of mass gap (universal, from crystal positivity) and confinement (specific, from C(K) = 0) is established: mass gap requires only P1, while confinement additionally requires zero commutation deficit. Complete classification tables for all types are provided. Zero free parameters per type. Keywords: Yang–Mills mass gap, Kashiwara crystal, compact simple Lie groups, Weyl group, commutation deficit, spectral convergence, zero free parameters, Lean 4