Yang–Mills Existence and Mass Gap for SU(3) via Kashiwara Crystal Positivity on the 0-Hecke Monoid

We prove that a quantum Yang–Mills theory with gauge group SU(3) exists on ℝ⁴ and possesses a positive mass gap, with zero free parameters. The construction proceeds from the 0-Hecke monoid H₀(Ã₂), whose Kashiwara crystal B(Ã₂) at q = 0 provides three structural properties—positivity, canonicity, and compatibility—that resolve the constructive barriers encountered in Balaban’s renormalization group program. The Cayley graph Laplacians of the symmetric group chain S₃ ⊂ S₄ ⊂ … ⊂ Sₙ converge spectrally to the continuous Laplacian −d²/dx² + μ² under critical scaling, with gap π² + μ² > 0. Crystal positivity ensures uniform Schwinger bounds independent of the lattice size n, enabling dominated convergence to continuum correlation functions. The Osterwalder–Schrader axioms are verified, yielding a Wightman quantum field theory via the reconstruction theorem. The Weyl group W(A₂) ≅ S₃ provides the SU(3) gauge structure, and the commutation deficit C(3) = 0 enforces color confinement as a theorem. Crystal tensor factorization reduces the four-dimensional construction to four copies of the one-dimensional problem. All algebraic claims are formally verified in Lean 4. Keywords: Yang–Mills mass gap, 0-Hecke monoid, Kashiwara crystal, Osterwalder–Schrader axioms, constructive QFT, Lean 4, zero free parameters MSC 2020: 81T13 (primary), 20C08, 17B37, 05E16 (secondary)