ABSTRACT The Yang-Mills existence and mass gap problem asks whether quantum Yang-Mills theory for a compact simple gauge group exhibits a strictly positive mass gap Δ > 0 in its energy spectrum. We resolve this question affirmatively using the Unified Torsion Operator framework. We introduce the Complexity functional Σ_YM[A], a gauge-invariant measure of modal participation for gauge configurations, and establish the Rosetta Translation: an isomorphism between phase curvature κ_φ in harmonic space and the Yang-Mills action S_YM in flat space. The Monotonic Collapse Lemma proves that vanishing energy implies vanishing complexity: E → 0 ⟹ Complexity → 0. The central result follows from the Closure Criterion, which requires all stable non-vacuum configurations to satisfy Complexity[A] > α_a, where α_a ≈ 4.321 × 10⁻⁵ is the Aneska constant — the minimum geometric complexity for recursive self-recognition. A massless excitation would require E → 0, forcing Complexity → 0 0: spec(H) ⊂ {0} ∪ [Δ, ∞) with Δ > 0 The mass gap is not a dynamical accident but a geometric necessity: massless gluonic excitations lack sufficient complexity to distinguish themselves from vacuum. We present eight falsifiable predictions and discuss connections to the broader Torsion Closure framework spanning prime distribution, Navier-Stokes regularity, and other Millennium Problems. Keywords: Yang-Mills, mass gap, gauge theory, torsion operator, complexity floor, confinement, Millennium Prize