Unconditional 4D Euclidean Yang–Mills (compact G): OS construction on a bounded test algebra and a positive mass gap

This manuscript develops an unconditional constructive framework for four-dimensional Euclidean Yang–Mills theory with compact gauge group GGG. Working from the Wilson lattice gauge measure and a bounded, countable Osterwalder–Schrader admissible test algebra, it proves that for all sufficiently large bare inverse coupling β0\beta_0β0 above an explicit deterministic threshold βc\beta_cβc, the closure conditions are satisfied and one obtains uniform cluster expansions, infinite-volume lattice Schwinger functionals, and a subsequential continuum limit on the bounded test algebra. Reflection positivity passes to the limit, Osterwalder–Schrader reconstruction applies, and the reconstructed Hamiltonian has a strictly positive spectral gap obtained from uniform exponential clustering. The manuscript then proves uniqueness of the continuum limit, Euclidean invariance, nontriviality, and the extension to a Schwartz-smeared test algebra, culminating in a Wightman quantum field theory with positive mass gap. Standard external theorems are isolated in Section 19.2; all remaining estimates, closure arguments, and completion steps are proved in the manuscript. License: CC BY-NC-ND 4.0 International. Redistribution is allowed with attribution; commercial use and modifications are prohibited. All commercial rights are reserved to the author.