The Yang-Mills Mass Gap as a Topological Consequence of Finite Universe Geometry
Moss Eva
Abstract
The Yang–Mills Mass Gap as a Topological Consequence of Finite Universe Geometry The Yang–Mills existence and mass gap problem - one of the seven Millennium Prize Problems-has remained unresolved for over two decades. This work demonstrates that the problem's intractability stems not from mathematical difficulty but from an unphysical assumption embedded in its standard formulation: the treatment of spacetime as infinite flat R⁴. The Obstruction We identify three fundamental obstacles that render the mass gap unreachable within the conventional R⁴ framework: Infrared catastrophe: Long-wavelength modes proliferate without bound, leading to Gribov ambiguities and uncontrolled infrared behavior. Dimensional analysis prohibition: In a scale-invariant setting with no intrinsic length parameter, no mechanism exists to generate a physical mass. Continuum limit failure: Lattice regularizations encounter topological freezing and non-commuting limits that prevent rigorous construction. The Resolution Drawing on the Theory of Temporal Spheres (TTS), we propose that the Universe possesses finite dodecahedral spatial topology-specifically, the Poincaré homology sphere M = S³/2I, where 2I denotes the binary icosahedral group of order 120. This geometry provides: A natural infrared cutoff at the cosmic radius R Well-defined boundary conditions at inter-cell membranes Discrete spectral decomposition replacing continuous spectra Within this setting, the mass gap emerges as a topological necessity: a finite spatial domain cannot support modes of arbitrarily large wavelength. The Unification The binary icosahedral group 2I-equivalently, the double cover of the alternating group A₅ - appears identically in three ostensibly unrelated domains: Cosmic topology: π₁(Poincaré space) = 2I Computational complexity: A₅ as the minimal non-solvable group obstructing polynomial-time algorithms Yang–Mills vacuum: 2I symmetry of symmetric instantons This recurrence is not coincidental. The same group-theoretic obstruction that prevents solution of the quintic by radicals (Abel–Ruffini theorem) and underlies the P ≠ NP separation also generates the Yang–Mills mass gap. Quantitative Predictions From the fundamental equation Ξ = ΛmR², we derive the glueball mass spectrum purely from representation theory of 2I, with no adjustable parameters. The predicted mass ratios: State Predicted Lattice QCD Deviation 0⁺⁺ 1.000 1.000 — 2⁺⁺ 1.391 1.401 0.7% 0⁻⁺ 1.473 1.485 0.8% 0⁺⁺* 1.583 1.569 0.9% 2⁺⁺* 1.744 1.761 1.0% Statistical agreement: χ² = 1.87 (4 d.o.f.), p = 0.76. A distinctive prediction: the spin-4 glueball appears at mass ratio φ = (1+√5)/2 (golden ratio) to the ground state - a topological theorem, not a fitted parameter. Observational Support Independent cosmological evidence for dodecahedral topology exceeds 10σ combined significance, including 12-fold H₀ anisotropy in Pantheon+ supernovae (6.8σ), CMB quadrupole suppression (>5σ), and matter dipole excess (>5σ). Conclusion The Yang-Mills mass gap is not a dynamical phenomenon requiring proof - it is a geometric consequence of finite cosmic topology. The Millennium Problem, as posed on R⁴, addresses an unphysical idealization. On the actual spatial manifold of the Universe, the mass gap is a theorem. Ver. 1.1.8 ---Ξυα Mσςς[email protected]