Environment-Dependent Weak-Lensing Residuals in KiDS-1000: A Bridge between Differential Shear and Conformal Response Theory

We present a differential weak-lensing analysis of environment-dependent shear residuals in KiDS DR4, cross-matched with the GAMA group catalogue, designed to bridge observational residual shapes and conformal response theory. By splitting lenses into field galaxies and group members with richness N_{\rm fof}\ge 10, we construct a shape-only differential tangential shear profile, \Delta\Delta\gamma_t^{\rm shape}(R), after projecting out the constant amplitude mode with the full jackknife covariance. The combined TOMO1–3 signal exhibits a characteristic inner depression with outer relaxation toward zero. We show that this morphology is well described by a minimal Hernquist-regularised two-scale response template motivated by conformal no-slip scalar–tensor theory, in which the high-environment selection relaxes more rapidly to the GR limit than the low-environment (field) selection. The full-covariance bridge fit yields \lambda_{\rm low}=2.18', \lambda_{\rm high}=13.41', a=3.31', and A=0.026, with \chi^2=14.06 for 14 radial bins and four free parameters. Beyond the baseline bridge fit, we identify an intermediate potential-depth quantile (Q3 in \log_{10}(M_A/R_{50})) whose tangential residual reaches \sim 3.7\sigma in TOMO2, persists under an even/odd patch split-half test, and passes the cross-shear null test (p=0.31). A direct comparison of Q3 against the pooled remaining quantiles yields a clean cross-shear null (p=0.84) with only moderate tangential separation, motivating targeted follow-up rather than a claim of a universal monotonic environment dependence. Pure mass-screening models with \lambda_{\rm high}<\lambda_{\rm low} do not produce stable physical solutions: their nominal improvements rely on bound-hitting at unphysical parameter values and disappear once conservative lower bounds are imposed. The stricter N_{\rm fof}\ge 20 selection is statistics-limited and consistent with null. Taken together, the results provide a controlled empirical bridge between environment-differential shear residuals and minimally conformal response models, while remaining explicitly conservative about interpretation.