Topological Obstruction to the Yang-Mills Mass Gap on ℝ⁴ and Geometric Resolution via Compact Constructible Manifolds

We demonstrate that the Clay Mathematics Institute's formulation of the Yang-Mills Mass Gap Millennium Problem is fundamentally ill-posed. In a concise, rigorous proof, we establish that the problem's dual requirements—satisfying the Osterwalder-Schrader (OS) axioms on a continuous ℝ⁴ manifold while exhibiting a strict mass gap (Δ > 0)—are topologically mutually exclusive.

The 1-Page Impossibility Theorem (The "No-Go" Proof):

The fundamental obstruction on continuous ℝ⁴ is proven through a strict topological chain. Using Singer’s theorem (1978) on the non-existence of global gauge sections, we show that any consistent non-abelian gauge theory on pure ℝ⁴ requires Gribov-Zwanziger infrared regularization to handle Gribov copies. This regularization inevitably introduces complex-conjugate poles into the gluon propagator, which strictly violates the OS axiom of Reflection Positivity. Therefore, the problem, exactly as formulated by the CMI, has no mathematical solution.

The Geometric Resolution:

Moving beyond the impossibility theorem, we provide the exact mathematical framework required to resolve the paradox. The mass gap is not an artifact of continuous quantum anomalies, but a strict geometric consequence of the vacuum's discrete algebraic topology.

By transitioning the base spacetime from ℝ⁴ to a compact constructible manifold $\mathcal{M} = S^4 \times \mathbb{T}^3$ over the multiquadratic field 𝕂 = ℚ(√2, √3, √5), the topological contradictions vanish. Within this framework:

• Geometric Mass Gap: The spectral gap emerges naturally as a non-zero lower bound via Cheeger's inequality ($\lambda_1 \geq h^2/4 > 0$).

• Topological Confinement: Color confinement is strictly enforced by the non-abelian topology of the braid group B₃, protected by the prime torus knot T(103,3) with a Seifert genus of g = 102.

The solution to the Mass Gap lies not in continuous perturbative models, but in the exact discrete geometry of the constructible vacuum.

Visual Documentation:
This work includes comprehensive mathematical visualizations demonstrating:
- The topological contradiction chain (Singer → Gribov copies → GZ regularization → OS violation)
- Spectral gap comparison between ℝ⁴ and S⁴×T³
- Braid group topology (B₂ vs B₃)
- Seifert surface structure for T(103,3)
- Constructible field hierarchy K = Q(√2, √3, √5)

Related Work:
See companion papers on the IT³ Framework for applications to MOND scale, solar cycle prediction, and W-boson mass precision.