Fixed-epsilon CutOut in the quasi-abelian class of the non-abelian Henstock–Kurzweil functional integral: direct comparison, oscillation, and the classification of regimes

This is the fourteenth 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. It attacks the question left open by the previous work: does the normalized CutOut ratio r_N(epsilon) of the quasi-abelian two-mode class of SU(2) on T^4 tend to zero at a fixed coupling epsilon > 0? All results of the previous works are used as black boxes and none is re-proved. The question is not resolved here, and none of the three possible outcomes is asserted; what the paper establishes is a precise description of the boundary of the present methods, together with two negative results which close two of the three proposed routes. (I) The monotonicity mechanism of the direct-comparison route is impossible: for a continuous perturbation weight h_N and an open Gribov region meeting the box B_R without filling it, one always has infB_R cap Omega h_N <= supB_R setminus Omega h_N, because the two regions share a boundary point interior to the box; the proposed suppression factor exp(-epsilon^2 [inf_Omega h_N - supB setminus Omega h_N]) is therefore always >= 1. (II) The premise of the oscillation route is false: the unperturbed truncated determinant does not alternate in sign, since the retained modes come in pairs {n, -n} and each factor is even in n, whence Delta_0(N)(A) >= 0 for all A and N; a sign-reversing involution acting through the determinant can exist only on the null set {Delta_0(N) = 0}. Two conditional criteria of that route survive and are proved in full: an exact measure-preserving sign-reversing involution makes the numerator vanish, and a polynomially decaying numerator with a denominator bounded below gives the CutOut. (III) The estimates (K4)–(K7) of the previous works do not decide the question: for every admissible tuple of constants there are data satisfying all of them with r_N -> 0, data with r_N bounded away from 0, and data with r_N unbounded. In particular the obstruction of the previous work — the failure of the error budget to be o(N) — is compatible with the CutOut, which settles the fallback statement (ii) of the programme; the fallback statement (i) is proved in conditional form only. Every theorem is proved in full in the main text and machine-verified in Lean 4 with no sorry, no admit and no new axiom, and a deterministic numerical script illustrates the two negative results at small N.