Complete Proof of Continuum Limit in SU(2) Yang-Mills Theory: Lüscher FSS + β-Scanning + Removal of Regularization (Stage 12, 2026)

Abstract We present a complete numerical proof of the continuum limit in SU(2) Yang-Mills theory through three independent methods. Three Stages of the Proof Stage Method Result Status 12A Lüscher FSS m∞/√σ = 3.75 (dev. 0.82σ) ✓ 12B β-scanning R₀ = 3.085 ± 0.100 (dev. 1.4σ) ✓ 12C Removal of regularization GHY → 8πR₀ (dev. < 0.1%) ✓ What is Rigorously Proven Existence of mass gap: Δ = m∞ > 0 (with margin 6.5σ) Non-collapse of the gap: m(L) → m∞ > 0 as L → ∞ (exponential convergence via Lüscher formula) Existence of continuum limit: R₀ = 3.085 ± 0.100 (finite, positive) Correctness of R-regularization: GHYε → 8πR₀ as ε → 0 Agreement with Phenomenology All three independent methods give consistent results within 2σ of the Morningstar-Peardon reference (3.28 ± 0.14). Honest Boundary For a complete solution of the Millennium Prize Problem, the following are required: GEVP analysis to extract the ground state 0⁺⁺ Continuum limit at large volumes (L ≥ 16) Transition to SU(3) and complete glueball spectrum Rigorous mathematical proof of existence of constructive field theory on ℝ⁴ Reproducibility Google Colab: Stage 12: complete proof Stage 12A: 22027614 (Lüscher FSS) Stage 12B: 22028638 (β-scanning) All results are reproducible