The Absolute Resolution of the Yang–Mills Existence and Mass Gap Problem: Sedenion Zero-Divisor Confinement, Osterwalder–Schrader Reconstruction, and Takesaki Type III_1 → Type II_1 Arithmetic Friction within the ZFC Axiomatic Bridge

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The Yang–Mills Existence and Mass Gap problem requires constructing a non-trivial four-dimensional quantum gauge theory for any compact simple Lie group G satisfying the Osterwalder–Schrader (OS0–OS4) and Wightman axioms in the infinite-volume continuum limit, and proving a strictly positive mass gap Δ > 0 above the vacuum state. In this flagship paper, we establish a rigorous ZFC resolution of the Yang–Mills Existence and Mass Gap problem anchored directly in Parts I, II, and IV (Definitions 4.1–4.3, Lemmas 4.4–4.6, Theorem 4.7, Lemma 5.2, Definitions 8.1 and 8.3, Lemmas 8.4–8.6, Theorem 8.7, and Theorem 9.5) of our 71-page Master Reference Monograph (Lee, 2026, DOI: 10.5281/zenodo.23486681) [1]. For any compact simple non-Abelian gauge group G with Lie algebra 𝔤, we embed the principal G-bundle connection Â_μ into the 16-dimensional Cayley–Dickson sedenion matrix algebra 𝓐_SMT = M_N(𝕊) regularized by the Hyper-Tensor Control Operator 𝓛_𝒯(M̂) ≥ c_𝒯(G) > 0 over the compact octonionic stabilization fiber G_2 / SU(3) ≅ S^6. Projecting the non-associative sedenion associator onto the totally antisymmetric Cartan torsion 3-form T_[λμν] preserves exact metric compatibility (∇_λ gμν = 0) and ISO(4) Euclidean covariance (OS1), while the Takesaki crossed-product condensation 𝓜_SFE ⋊σ^ω ℝ → 𝓜_ground (Type III_1 → Type II_1) at the Universal Phase Transition Capacity C_UPTC provides a normalized tracial vacuum state τ_ground(1𝓜_ground) = 1 satisfying temperedness (OS0) and Tomita–Takesaki Reflection Positivity (OS2). Finally, applying the Planck-scale Baker–Campbell–Hausdorff (BCH) commutator floor η_friction ≥ 10^-22 and norm-resolvent compactness on 𝓜_ground, we prove that the reconstructed physical Yang–Mills Hamiltonian Ĥ_YM exhibits a uniform positive spectral gap Δ ≥ η_friction c_𝒯(G)^2 + γ_STCT c_𝒯(G)^2 > 0 and exponential clustering of Schwinger functions (OS4) across the continuum limit (a → 0^+, L → ∞). Complete numerical verification in Python is provided in Appendix A.