Gribov Copies and Gauge Invariance: A Henstock–Kurzweil Continuum Integral on the SU(3) Lattice

The Feynman functional integral of a non-Abelian gauge theory has a Gribov ambiguity: the Landau gauge condition has several solutions on one orbit, the Gribov copies. The gauge-fixed integral is usually defined by restriction to a surface, corrected by the Faddeev–Popov determinant. Gribov showed in 1978 that no such surface exists globally: the integral is not uniquely defined. The method adopted here is the Henstock–Kurzweil (HK) continuum integral on Kuelbs–Steadman spaces: non-absolute, needing neither Wick rotation nor a σ-additive measure. In the present paper it is applied to SU(3) gauge theory on the periodic lattice {0,1}^4. The main result concerns gauge-invariant functionals: the HK Riemann sums built on two different gauge surfaces are equal term by term, so that the cancellation is established at the level of the approximating sums rather than of their limits, and the copies can affect only gauge-variant quantities. The finiteness statement for copies counted as gauge transformations is false; two corrected statements replace it. The paper is a methodological demonstration rather than a new physical claim: the cancellation of the Gribov ambiguity for gauge-invariant functionals was never in doubt on physical grounds, but its formulation as a term-by-term identity in a non-absolute integral is new. In the traditional Faddeev–Popov framework the functional integral is a formal symbol; here it is a constructed object, and every statement about it is proved from the definitions. A deterministic computation exhibits a configuration with two copies and confirms this to 2.4 × 10⁻¹³.