Fixed-epsilon CutOut in the quasi-abelian class of the non-abelian Henstock–Kurzweil functional integral: direct comparison, oscillation, and the classification of regimes
Abstract
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 inf_{B_R cap Omega} h_N <= sup_{B_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 - sup_{B 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.
Related
- 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
- 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
- Algebraic Completeness of Feynman Premeasures: The Henstock-Kurzweil Path Integral and Operator Wick Calculus in Lean 4
- Henstock--Kurzweil Gauge Integral in the Non--Gaussian Regime: A Machine--Verified Construction
- Gribov Copies and Gauge Invariance: A Henstock–Kurzweil Continuum Integral on the SU(3) Lattice
BibTeX
@misc{berdinsky2026fixedepsilon,
title = {{Fixed-epsilon CutOut in the quasi-abelian class of the non-abelian Henstock–Kurzweil functional integral: direct comparison, oscillation, and the classification of regimes}},
author = {Yu. N. Berdinsky},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23578376},
url = {https://zenodo.org/records/23578376},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fixed-epsilon CutOut in the quasi-abelian class of the non-abelian Henstock–Kurzweil functional integral: direct comparison, oscillation, and the classification of regimes
AU - Yu. N. Berdinsky
PY - 2026
DA - 2026/10/22
PB - Zenodo
DO - 10.5281/zenodo.23578376
UR - https://zenodo.org/records/23578376
AB - 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 inf_{B_R cap Omega} h_N <= sup_{B_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 - sup_{B 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.
ER -