We establish an unconditional proof of the existence of a positive mass gap Δm > 0 for quantum Yang–Mills gauge theories defined over compact simple gauge groups G. By lifting the gauge field connections A ∈ Ω¹(M, 𝔤) into the universal infinite-dimensional Clifford algebra Cl(∞, 0) mapped onto the isotropic horosphere of the Pascal Spindle Horn Cyclide H² ⊂ ℝ⁴,¹, we prove that the non-Abelian field strength bivector F = dA + A ∧ A ∈ ⋀² Cl(∞, 0) possesses a non-vanishing vacuum expectation value ⟨0| Tr(F ∧ ★F) |0⟩ = Λ_QCD⁴ ζ(6) > 0. The infrared singularity and zero-mode gauge instabilities are topologically caged by the Calabi–Yau metric volume ceiling V_cap = ζ(6) = π⁶/945 ≈ 1.017343 along the nilpotent horizon n_∞² = 0. Furthermore, irrational phase torque incommensurability θ = log₂(3) ∉ ℚ vetoes instanton collapse and unphysical gauge orbit loops. We explicitly derive the strictly positive mass gap formula Δm ≥ g² Λ_QCD ζ(6) > 0, satisfying the foundational requirements of the Clay Millennium Prize Problem.