Normalized CutOut in the Non-Abelian Continuum Henstock–Kurzweil Integral: The Cartan Class of Configurations and zeta-Regularization of the Faddeev–Popov Determinant

This is the tenth paper of a programme constructing the Feynman functional integral without a Wick rotation, based on the non-absolutely convergent Henstock–Kurzweil (HK) integral and on Kuelbs–Steadman spaces. Its predecessor formalized the finite-mode non-abelian continuum and left the CutOut problem — the vanishing, in the continuum limit, of the contribution of the configurations lying outside the first Gribov region — open, proving only a conditional statement. The present paper attacks the CutOut problem in the form in which it is meaningful: not the absolute statement, which is false because the bare truncated Faddeev–Popov determinant diverges with the truncation level, but the normalized statement, in which the contribution of the exterior of the Gribov region is divided by the full integral over the box. For the constant Cartan class of SU(2) on the four-torus, where the spectrum of the Faddeev–Popov operator is known explicitly, the normalized statement is proved: for every Fresnel coefficient γ > 0 and every box B_R with R ≥ π, the normalized CutOut ratio tends to zero, and is bounded by C(2N+1)^8 e^{-N} with a constant C = C(R) depending only on the box. Importantly, the Yang–Mills action on the constant Cartan class is identically zero, so the theorem does not make any claim about a dynamical mechanism of cut-out; it proves only that the normalized Faddeev–Popov determinant concentrates in the first Gribov region and that the Fresnel weight of the premeasure does not obstruct this. The singular-direction hypothesis of the previous work is also proved for the Cartan class with constants independent of the truncation level. The Seeley ζ-regularization is treated at paper level, with the finite-mode identity det_ζ = exp(-ζ'(0)) proved; the analytic continuation of the continuum spectral sum is not formalized, and this is stated explicitly. All the statements of Sections 3–7 that enter the proofs of Theorems A and B are machine-verified in Lean 4 with no sorry, no admit and no new axioms; a deterministic numerical script, using two independent quadrature methods and a high-precision cross-check, confirms the estimates on the one-dimensional slice.