Degeneracies between Metric Deformations and Thin-Disk Models in Black Hole Images: A Channel-by-Channel Analysis

We present a systematic study of how modifications of the inverse Kerr metric manifest in images of a thin accretion disk and assess the sensitivity of different image observables to these deformations within a fixed astrophysical model. Using backward ray tracing in Boyer--Lindquist coordinates, we introduce three one-parameter deformations of the inverse metric---in g^{tt}, g^{rr}, and g^{\phi\phi}---of the form g^{\mu\nu} -> g^{\mu\nu}[1 + \epsilon (M/r)^2], with |\epsilon| <= 0.1. For each modification we construct a thin-disk image with a power-law emission profile I_em \propto (r_in/r)^3 (motivated by the Novikov--Thorne profile) for spin a \in {0, 0.5, 0.9, 0.99} and inclination i \in {30^\circ, 70^\circ, 80^\circ, 85^\circ} on the main grid n_pix = 80, with verification on n_pix = 120 and 160, and extract five observables: shadow radius R_sh, centroid shift, brightness asymmetry A_asym, peak brightness I_peak, and integrated brightness I_tot. We find: (i) the g^{tt} modification shifts both the shadow boundary and the photometric peak; (ii) the g^{rr} modification is strictly invisible in the shadow geometry over the entire grid (an analytical consequence of the structure of the null geodesic equations), and its photometric signal is numerically undetectable at realistic accuracy; (iii) the g^{\phi\phi} modification produces an intermediate photometric signal with opposite sign relative to g^{tt}. We additionally performed a stability analysis of the observables against grid resolution: the shadow radius R_sh (computed by the capture-boundary bisection method with search step \pm 0.005 M and residual systematics \sim 0.5\%) is found to converge, whereas the peak and integrated disk brightnesses exhibit systematic non-convergence of the derivatives with respect to \epsilon, up to a sign change between grids n_pix = 80 and 160. Restricting the Fisher analysis to the stable observable R_sh, we obtain \sigma_\epsilon \approx 0.43 (g^{tt}, a = 0.9), \approx 0.25 (g^{tt}, a = 0.99), \approx 0.68 (g^{\phi\phi}, a = 0.99); for g^{rr} the derivative dR_sh/d\epsilon vanishes identically, rendering \sigma_\epsilon formally infinite. We analytically show that the invariance of the critical curve with respect to g^{rr} is a strict consequence of the structure of the null geodesic equations; numerically, this is confirmed across all three grids and by the bisection method with accuracy 10^{-4} M in the derivative dR_sh/d\epsilon. Implications for testing modified theories of gravity with next-generation observations and the limitations imposed by degeneracies of a single disk are discussed.