A Geometric Program for Yang-Mills Configuration Spaces: From the Mass Gap to Macroscopic QCD

We develop a geometric framework for the analysis of the Yang-Mills configuration space \mathcal{B} = \mathcal{A}/\mathcal{G}, the infinite-dimensional quotient of the Sobolev space of connections by the gauge group. The projection \pi: \mathcal{A} \to \mathcal{B} is a Riemannian submersion whose curvature is controlled by the geometry of gauge orbits. Main structural result is a mean curvature identity: under a zeta-regularisation of the Faddeev-Popov determinant and within the first Gribov region, the mean curvature vector of each gauge orbit equals the logarithmic gradient of (\det \Delta_{FP})^{1/2}. This yields an explicit formula for the Bakry-Émery Ricci tensor of \mathcal{B}, identifying ghost fields geometrically as curvature contributions from fibre volume variation. We prove that the Gribov horizon is a curvature singularity: every finite-length horizontal curve approaching it accumulates infinite mean curvature, and the Parisi-Wu stochastic flow is confined to \Omega_0 almost surely for all finite times. Formulate a curvature-dimension conjecture CD(\rho,\infty) for the configuration space in four dimensions. Under this conjecture, we derive the full geometric chain connecting the Hausdorff dimension of flux tubes to the Yang-Mills spectral gap via isoperimetric bounds, optimal transport, and Log-Sobolev inequalities. The geometric mechanism is extended to finite-temperature QCD, proving that topological melting leads to simultaneous deconfinement and chiral symmetry restoration at T_c. Furthermore, the Chiral Magnetic Effect is recast non-perturbatively as a Wasserstein gradient flow of the Dirac spectral density. The entire program is rigorously verified in two-dimensional Yang-Mills theory and the Abelian case.