Exceptional Algebraic Regularization of Four-Dimensional Gauge Connections and the Infrared Mass Gap Floor

We present a rigorous constructive mathematical framework for quantum Yang--Mills theory on standard four-dimensional domains (M = ℝ4 and M = 𝕋4) establishing the existence of a strictly positive mass gap Δ > 0 for compact simple gauge groups. To resolve the ultraviolet divergences inherent in continuous point-like manifolds without introducing ad-hoc lattices or violating gauge covariance, we map the classical gauge fields via a full and faithful covariant embedding into a dual spectral triple (A, Hphys, DExc). The internal algebra is constructed as the tensor product of smooth manifold functions and the 27-dimensional complexified exceptional Jordan algebra J3(𝕆 ⊗ ℂ). Its octonionic realization and Rosenfeld-projective interpretation are described in the cited literature. At the trans-Planckian limit (x → 0), extreme convective concentration activates the non-vanishing matrix associator [X, Y, Z] ≠ 0 of the underlying octonionic coordinate ring. Under the Weitzenböck--Lichnerowicz decomposition of the squared Exceptional Dirac Operator, this non-associative obstruction transmutes into a rigid, non-local potential floor Tr(ΦAssoc) ≥ 3/8 > 0. Crucially, the Platonic shadow projection Yang-Mills Lagrangian remains strictly unmodified on the real manifold, and the background coupling parameter gG is preserved. The regularizing potential arises solely as an operator-valued representation of the flow in the dual spectral category, acting as an analytical device dually analogous to complex contour integration. We prove that any potential singularity carries exactly zero norm inside the Gelfand--Naimark--Segal (GNS) null ideal N, cleanly recovering strict Wightman locality and satisfying the Osterwalder--Schrader reflection positivity axioms while enforcing a non-zero infrared mass floor. Note: The digital integrity records for this monograph, alongside the synchronized master assets of the Project Solitude ecosystem, are recorded in the accompanying file: cryptographic_manifest.txt.