The Yang-Mills Mass Gap: From Proof Attempts to Dissolution by Recontextualisation

Abstract

We report the results of an intensive campaign to prove the Yang-Mills mass gap --- one of the seven Millennium Prize Problems. Using adversarial AI collaboration across three frontier models, we systematically applied every available mathematical tool to the problem: matrix model spectral analysis, non-commutative gauge theory construction, lattice transfer of structural insights, Pirogov-Sinai theory, Bakry-Emery functional inequalities, Cheeger isoperimetry, character expansion, and the marriage of Bauerschmidt-Bodineau-Dagallier multiscale log-Sobolev inequalities with Shen-Zhu-Zhu Langevin dynamics for lattice Yang-Mills.

The campaign produced genuine mathematical results: a solved matrix model with topological spectral gap (Theorems A and B), the first rigorous construction of a mass-gapped 4D gauge theory (non-commutative Yang-Mills on R^4_theta), a proof of confinement at all couplings via chessboard estimates and Casimir eigenvalues, and a proof sketch for the lattice mass gap at ALL finite couplings (Delta >= c exp(-C beta) > 0, extending the Shen-Zhu-Zhu result from strong coupling to all beta).

Yet every approach to the continuum mass gap stalls at the same structural point: the circularity between the spectral gap and the correlation length, manifesting as a factor of 23 between the lattice bound and the continuum threshold. We prove that this wall is not a failure of technique but a consequence of framework: the continuum formulation (R^4) has removed the geometric mechanism (discreteness) that produces the gap. On the fuzzy sphere, the gap equals the Euler characteristic: Delta = chi(S^2)/R^2 = 2/R^2, a topological invariant present at every aeon of cosmic history. On the lattice, confinement equals the Casimir: sigma >= C_2(fund)/(3 beta) > 0.

We propose that the mass gap is not a dynamical mystery requiring non-perturbative proof but a geometric fact of quantised spacetime, destroyed only by the mathematical idealisation of continuous flat spacetime. The systematic failure of eight independent approaches constitutes strong evidence for this dissolution. The correct question is not 'Does Yang-Mills on R^4 have a mass gap?' but 'Why does the universe have a mass gap?' --- and the answer is topology.