Normalized CutOut for the quasi-abelian two-mode class in the non-abelian continuum Henstock–Kurzweil integral: banded Faddeev–Popov operator and numerical comparison with lattice data

This is the twelfth paper of a programme constructing the Feynman functional integral without a Wick rotation on the basis of the non-absolutely convergent Henstock–Kurzweil (HK) integral and Kuelbs–Steadman spaces. The previous works proved the normalized continuum CutOut for the constant Cartan class of SU(2) on the four-torus and for the constant non-Cartan class; in both cases the configuration is constant in space-time, so that the Faddeev–Popov operator is block diagonal in the Fourier modes. The present paper treats the first class with a genuinely x-dependent Faddeev–Popov operator: the quasi-abelian two-mode class A_mu(x) = A_mu + epsilon (Ahat_mu exp(2 pi i k·x) + Ahat_mu^* exp(-2 pi i k·x)), with A in su(2)^4 = R^12 the integration variable, a fixed mode k in Z^4 \ {0}, and fixed transverse amplitudes Ahat. Target B is achieved, Target A is not. Proved here: the truncated Faddeev–Popov matrix of the class is hermitian, hence has a real determinant, exactly under the transversality condition k_mu Ahat_mu = 0; it is a banded block matrix whose band-width is bounded by |k|_infinity independently of the truncation level N, and it is block tridiagonal along the decoupled chains n + Z k of the mode lattice; the class reduces to the constant class of the previous work at epsilon = 0 and, with a shifted configuration, at k = 0; the first Gribov region of the class, defined intrinsically by positive definiteness, satisfies the outer bound OmegaN,k subset {|A_mu| < 2 pi} and the inner bound B_c subset OmegaN,k for c <= pi/(4 sqrt 3) and epsilon ||Ahat||_1 <= pi/(4 sqrt 2), both uniform in N; a Schur complement with respect to the N-independent set of exceptional modes reduces the determinant to a product of a concentrating factor and a perturbative factor det(1 + epsilon K) whose logarithm obeys |log |det(1 + epsilon K)|| <= eta_N = C(epsilon^2 N^2 + epsilon^4 N^3), the odd orders of the expansion vanishing because a closed walk of ±k steps has even length; and the normalized CutOut limit holds whenever sqrt N epsilon_N -> 0, in particular for epsilon_N = N⁻ᵈᵉˡᵗᵃ with delta > 1/2, with the explicit rate C(2N+1)^24 exp(-bN + 2 eta_N), which is exponential in N. For fixed epsilon the second-order term is of size N^2 and the argument does not close; Target A is therefore stated as a conjecture and is not proved, and the optional Target D is consequently not included. A separate section compares the density of near-zero Faddeev–Popov modes of the class with qualitative features reported in the lattice literature; it is a discussion, not a theorem, it makes no claim of quantitative agreement, and it states openly one feature in which the present truncation differs from the lattice situation. Every theorem is proved in full in the main text; the structural statements and the assembly are additionally machine-verified in Lean 4 with no sorry, no admit and no new axioms, and a deterministic numerical script confirms the structure and the estimates at N <= 4.