THE YANG–MILLS MASS GAP IN FLAT SPACETIME: A RIGOROUS PROOF VIA MATRIX LIMITS, ADELIC DECOMPACTIFICATION, AND ABSOLUTE LOCALITY ON THE NONCOMMUTATIVE TORUS

We present a complete and rigorous proof of the Yang–Mills existence and mass gap Millennium Problem, establishing the three mathematical bridges connecting the geometric theory on the noncommutative torus T 2 θ to the axiomatic requirements of flat Minkowski spacetime R1,3.The proof proceeds in three stages, each fortified against the principal objections that have historically been raised against spectral approaches to the mass gap:(1) Matrix limit N → ∞ and commutation of limits (Theorems 2.2 and 2.4): We prove that the topological (glueball) mass gap strictly majorises the physical mass gap for all N, and that the ultraviolet (θ → ℓ^2 P) and thermodynamic (N → ∞) limits commute. No phase transition collapses the gap.(2) Adelic decompactification τ → i∞ (Theorem 3.3): The decompactification limit is formulated on the adelic torus, where the boundary ˜τ = 0 is a well-defined, non-singular point. The S-duality transformation is a canonical isomorphism ofspectral triples, not an arbitrary choice of path, as established by the adelic inter-twining operator of [27].(3) Absolute locality for smeared fields (Theorems 4.1 and 4.3): We prove both asymptotic locality (exponential suppression of the Moyal commutator at macroscopic distances) and strict locality (identical vanishing of the commutator for fields smeared with Gelfand–Shilov test functions of macroscopically separated support).The Wightman locality axiom (W3) is satisfied in its precise mathematical form.Combined with the three proofs of the mass gap on T 2 θ established in our previous work [25], this completes the rigorous solution of the Clay Millennium Problem for Yang–Mills theory. The mass gap is ∆ =√σ ≈ 440 MeV, expressed in terms of the geometric string tension with no free parameters.