Sharpened CutOut and the fixed-epsilon boundary in the quasi-abelian class of the non-abelian Henstock–Kurzweil functional integral: optimal non-degeneracy radius, extension to finitely many modes, and loss of uniform control

This is the thirteenth 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 of Kuelbs–Steadman spaces, and it is the closing paper for the quasi-abelian class. The paper "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" established the normalized continuum CutOut for the quasi-abelian two-mode class of SU(2) on T^4 under the N-dependent force law epsilon_N = N^{-delta}, delta > 1/2, at the rate C(2N+1)^{24} exp(-bN+2eta_N). All results of that paper are used here as black boxes and none of them is re-proved. The present paper does three things. 

Part I (strengthening). The non-degeneracy radius rho_N = pi/(4(2N+1)^2) = Theta(N^{-2}) of the previous works, which came from the crude bound x_n <= 3 rho_N^2/pi^2 discarding the decay in |n|, is replaced by rho_N^* = pi/(4 sqrt(96)(2N+1)) = Theta(N^{-1}); the proof rests on the exact shell count #{|n|_inf = r} = 64r^3 + 16r and the resulting two-sided lattice estimate 8N(N+1) <= sum |n|^{-2} <= 32N(N+1) + 16N, and the scaling Theta(N^{-1}) is shown to be optimal by a matching lower bound at the corner configuration. Since the configuration space is su(2)^4 = R^12 and the assembly loses the volume factor (2 rho_N)^{-12}, the polynomial factor of the rate improves from (2N+1)^{24} to (2N+1)^{12}, the exponential rate exp(-bN+2eta_N) being unchanged. The whole CutOut statement is then extended from one perturbation mode to a finite set K={k_1,...,k_m}: the truncated Faddeev–Popov operator is banded with band width W_K = max_j |k_j|_inf independent of N, the retained mode lattice splits into the cosets of Lambda = <k_1,...,k_m>, the truncated determinant factorizes over that decomposition, and the budget keeps the scaling eta_N = C_* epsilon_N^2 N^2 + O(epsilon_N^4 N^2) with a constant that grows linearly in m, because the cross contributions of two distinct modes do not close.

Part II (obstruction). At fixed epsilon the method of the previous work cannot be made to close. The second-order term tr K^2 is a sum of squared Hilbert–Schmidt norms, one for each closed two-step path of the mode lattice, so no algebraic or phase cancellation exists; it satisfies c_0 N^2 <= tr K^2 <= c_1 N^2 with the explicit constants c_0 = ||Ahat||_2^2 / (16 pi^2 (1+W_K)^2) and c_1 = 80 ||Ahat||_1^2 / pi^2 for every non-zero transverse amplitude — no general-position hypothesis is needed, because the off-diagonal part of the quadratic form n -> |n Ahat|^2 is annihilated by the sign symmetry of the lattice. Consequently (1/N) tr K^2 -> infinity and no constant C satisfies tr K^2 <= CN: the error budget cannot be o(N) at fixed epsilon.

Part III (open problem and discussion). Whether the normalized CutOut ratio itself fails to tend to zero at fixed epsilon is stated as an open problem with three possible outcomes, none of which is asserted; a closing discussion describes three directions towards the global Gribov problem, each with an explicit statement of what is missing. Every theorem is proved in full in the main text; the statements are additionally machine-verified in Lean 4 with no sorry, no admit and no new axioms, and a deterministic numerical script confirms the identities and the scalings at small N.