PROJECT T Volume XL : The Continuum Limit Spectral Convergence, Critical Scaling, and the Indestructible Mass Gap

We prove that the Cayley graph Laplacians of the symmetric group chain S₃ ⊂ S₄ ⊂ … ⊂ Sₙ converge, under the rescaling L̃ₙ = n² · L(Sₙ), to the continuous Laplacian operator −d²/dx² on L²([0,1]). The spectral gap satisfies n²λ₁(Sₙ) → π² with convergence rate O(1/n²), and the k-th eigenvalue converges to (kπ)². Combined with the Mass Ladder Theorem (Vol. XXXII), this yields a massive Klein–Gordon operator −d²/dx² + μ² under critical scaling m = μ²/n², with continuum gap π² + μ² > 0. The Klein–Gordon identity Kₘ(t)/K₀(t) = exp(−mt) is proven to be an exact algebraic identity at every finite n, not an approximation. Correlation functions decay exponentially with rate √(m² + λ₁), confirming that the mass gap controls the correlation length. We establish that the spectral gap is indestructible: positive at every finite n, in the continuum limit, and in the strong-coupling limit. This volume opens the Clay Arc, building the bridge from discrete algebra to continuous quantum field theory with zero free parameters.