NHJQ-SK final project record (v3) The J-Q model under an imaginary Dzyaloshinskii–Moriya perturbation: mechanism, sign-free validation, the structure of the discrimination problem

We report the closure record of NHJQ-SK, an independent quantum Monte Carlo (QMC) study of the square-lattice J-Q model under a non-Hermitian perturbation H_nH = −iδJ Σ_⟨ij⟩ (S_i × S_j)_z — an imaginary Dzyaloshinskii–Moriya (DM) coupling. At the operator level the perturbation exactly reduces the symmetry SU(2) → U(1)_z. Exact-diagonalization (ED) anchors at L = 4×4 show a monotone, nonlinearly growing splitting between the in-plane and longitudinal staggered channels (ratio S^x²/S^z² from 1.000000 at δ = 0 to 1.004033 at δ = 0.9), consistent at the mechanism level with the easy-plane picture reported by Zou et al. for the non-Hermitian J-Q family. The model remains sign-free for |δ| ≤ 1 in the Marshall-rotated basis: the two spin-flip channels carry weights (J/2)(1 ± δ), so at the boundary δ = 1 one vertex weight degenerates exactly to zero and none becomes negative — verified empirically (sign ≡ 1.000, zero negative-sign fraction, in all 25 stored path-β δ = 1 runs and the 10 tasks of the longitudinal δ = 1 scan). We then document why the nature of the underlying Néel–VBS transition remains structurally underdetermined at the sizes accessible to this program: finite-size drift is expected under every scenario surviving our data; the correlation lengths of the competing scenarios can exceed accessible sizes by orders of magnitude; and the pre-registered first-order confirmators were not executed. Finally, we retract the headline claim of V2 — a ≈20σ "super-linear finite-size deviation" at δ = 1 presented as a thermodynamic signal — on grounds of evidential form alone: a bare single-momentum structure factor without any dimensionless crossing diagnostic cannot establish thermodynamic criticality at any coupling and in any channel. The dataset itself remains valid and is deposited in full. No verdict on the nature of the transition is rendered, and no critical-coupling value for δ > 0 is claimed.