Resolution of the Yang-Mills Mass Gap Problem via Srinivas Bounded Mathematics

We demonstrate that the Yang-Mills mass gap problem—one of the seven Clay Millennium Prize Problems—admits a natural resolution within the framework of Srinivas Bounded Mathematics (SBM). The mass gap problem asks whether a non-trivial interacting four-dimensional Yang-Mills gauge theory possesses a positive energy gap Δ > 0 between the vacuum and the lowest excited state. Traditional quantum field theory approaches face ultraviolet divergences requiring renormalization, leaving the rigorous existence of the mass gap uncertain. By applying Axiom 1 of SBM (Prohibition of Infinite Subdivision), which establishes that all spatial and momentum scales must satisfy r ≥ ε_AR > 0 for any finite positive AR Bound ε_AR, we show that: (1) loop integrals acquire natural ultraviolet cutoffs k_max = ℏ/(2ε_AR), rendering all quantum corrections finite; (2) the impossibility of infinite momentum subdivision prevents the accumulation of infinitely many massless modes required for Δ = 0; (3) the mass gap Δ > 0 emerges as a structural necessity rather than a dynamical accident. The resolution holds universally for any finite positive ε_AR > 0—the proof depends only on finiteness, not on specific value. As ε_AR → 0⁺, we have Δ → ∞, but crucially Δ > 0 always—the gap is never zero for any finite bound. Moreover, by the Non-Zero Measurability Theorem established in the SBM foundation: the mass gap Δ is a measurable physical quantity, and all measurable quantities satisfy |Q| ≥ ε_AR > 0; hence Δ > 0 follows immediately as a one-line consequence. The detailed proof structure in this paper provides the explicit construction and physical validation of this fundamental result. The framework provides rigorous mathematical foundation for what lattice QCD achieves computationally through finite discretization, aligns with the SBM-modified Wightman axioms by providing natural cutoffs for tempered distributions, and ensures Osterwalder-Schrader positivity via bounded operators. Version 7 updates: Aligned with SBM Foundation v3 (DOI: 10.5281/zenodo.18621608), incorporating the 0 vs ◦ (SBM Absolute Zero) distinction, Non-Zero Measurability Theorem perspective, Historical Analogy remark, updated AI disclosure, Sobolev constant citations (Talenti 1976, Aubin 1976), and SBM-modified Wightman/Osterwalder-Schrader axiom compliance.