ON THE MEASURE-ZERO PROPERTY OF THE GRIBOV HORIZON

A key step in the rigorous, non-perturbative construction of quantum Yang-Millstheory is the restriction of the functional integral to a gauge-fixed domain known as the Gribovregion Ω. For the resulting quantum theory to be well-defined, the boundary of this domain—theGribov horizon ∂Ω—must be a set of measure zero with respect to the functional measure onthe space of connections. This paper provides a complete proof of this foundational result. Westratify the horizon into sets where the Faddeev-Popov operator’s kernel has a fixed dimensionk. For the non-degenerate stratum (k = 1), we prove that the set of critical points of theeigenvalue map is empty, which implies this stratum has measure zero. For the degeneratestrata (k ≥ 2), we use the pre-image theorem to prove they are contained in finite-codimensionsubmanifolds, which also have measure zero. The core of the argument for the degenerate stratarests on an application of the maximum principle to prove the linear independence of certaineigenfunction commutators. This independence result is established via a Bochner-type identityand a rigorous maximum ratio argument, supported by a proof of the exponential decay of theoperator’s eigenfunctions.