From Wilson Blocks to a Yang–Mills Mass Gap: A Renormalization-Group Construction of the Four-Dimensional Observable Sector

We construct the gauge-invariant local observable sector of four dimensional Yang-Mills theory from Wilson’s lattice regularization along the asymptotically free bare trajectory. For every fixed compact connected simple gauge group, the finite-volume measures have a continuum Osterwalder-Schrader limit whose reconstructed local Hamiltonian has a strictly positive gap above the vacuum. The reconstructed sector is nontrivial: it contains a nonzero finite-energy local curvature excitation.The proof combines a short-distance OS construction of curvature polynomial fields with a retained-scale Balaban block-polymer expansion and a Kotecky-Preiss clustering estimate for separated bounded character sources. The source-comparison theorem identifies the relevant continuum local representatives with this source class, while a physical-scale local-channel clustering estimate supplies the exponential rate used by the OS semigroup criterion. The result concerns the cyclic gauge-invariant local observable sector constructed in the paper; it does not adjoin charged, topological, boundary-condition, or other complementary sectors. This paper is currently a working draft, comments welcome.