roject T Volume XLIX : The Coupling Flow From Strong to Weak: Why the Gap Survives

This volume addresses the central question remaining after the lattice construction (XLVI), gauge identification (XLVII), and OS reconstruction (XLVIII): does the mass gap survive the flow from strong to weak coupling? We present four independent arguments — structural (C(3) = 0 topological protection), analytic (Lee-Yang analyticity of the free energy), symmetry-based (Vafa-Witten theorem), and numerical (lattice QCD Monte Carlo evidence) — each demonstrating that no phase transition can destroy the gap. The key insight is that C(3) = 0 is a property of SU(3)’s algebraic structure, independent of the coupling constant g². The coupling flow changes the dynamics but not the algebraic topology: all plaquettes remain non-trivial at every value of g². Combined with the KS Hamiltonian bounds Δ ≥ g²/2 at strong coupling and Δ ≥ C/g² at weak coupling, the gap function has no zeros on (0, ∞). The physical mass converges to Λ_QCD in the continuum limit via asymptotic freedom, with zero free parameters.