The Ontological Paradox of the Yang-Mills Mass Gap: Topological Obstructions on the $\mathbb{R}^4$ Continuum
Year: 2026
Type: preprint
Abstract: The standard formulation of the Yang-Mills Millennium Prize Problem demands the rigorous establishment of a mass gap $\Delta > 0$ on the non-compact Euclidean spacetime $\mathbb{R}^4$. This paper demonstrates that this requirement constitutes an ontological paradox. Drawing a conceptual parallel to recent findings in causal topology, this work rigorously proves that the continuous, boundary-less geometry of $\mathbb{R}^4$ fundamentally obstructs the non-perturbative structure of non-abelian gauge fields. 💻 Interactive Lean 4 Environment (Zero sorry, Zero external axioms, compiles in real-time): File: RoundSquare.lean Site: https://live.lean-lang.org Through a systematic causal chain of axiomatic collapse, the manuscript formalizes four independent No-Go theorems proving that enforcing a mass gap on $\mathbb{R}^4$ inevitably violates core foundational axioms: The Axiomatic Obstruction: Restricting the path integral to the Gribov region to eliminate topological overcounting rigorously breaks nilpotent BRST symmetry. This explicitly violates the Osterwalder-Schrader Reflection Positivity axiom, destroying the positive-definite Källén-Lehmann spectral measure. The Geometric Obstruction: By Singer's Theorem, the lack of intrinsic compactness forces the functional measure into singular semi-algebraic traps (cusps) at the Gribov Horizon. This measure-theoretic void mathematically prevents the Hamiltonian from admitting a unique self-adjoint Friedrichs extension compatible with microscopic locality. The Algebraic Obstruction: Under the Haag-Kastler framework and the Atiyah-Singer index theorem, isolating a topological $\theta$-vacuum to secure a mass gap requires integration over the spatial boundary at infinity ($S^3_\infty$). By Strocchi's Theorem, this necessitates non-local projection operators that explicitly destroy Wightman microcausality. The Analytic Obstruction (The Cauchy Trap): The generation of a mass scale via dimensional transmutation introduces complex conjugate poles in the momentum space propagators. By Cauchy's Integral Theorem, these non-analyticities strictly obstruct Wick rotation, permanently severing the mathematical link between the Euclidean functional integral and the physical Minkowski spacetime. The Catch-22 of the Clay Formulation: One cannot generate a physical mass scale $\Lambda_{QCD}$ on a smooth, non-compact $\mathbb{R}^4$ without tearing apart the very axiomatic fabric (Wightman/OS axioms) required to define it. The problem cannot be solved as stated because it asks for a physically real phenomenon to occur within a mathematically incompatible reality. A rigorous resolution of quantum Yang-Mills theory requires a paradigm shift away from this sterile continuum toward discrete or compact topological manifolds. Files in this repository: Game_Over_CMI.pdf: The full academic manuscript. Game_Over_CMI.tex: The LaTeX source code of the manuscript. Game_Over_CMI_Picture.py: A reproducible Python script utilizing matplotlib to generate the exact analytic and geometric figures (Reflection Positivity violation, Spectral Density, Gribov Horizon cusps, and the Cauchy Trap) presented in the paper.
Author: Logvinovich Victor
Language: English
Subfield: Nuclear and High Energy Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7165366287