Probing the scalar hair of rotating Horndeski black holes through thick disk images

Horizon-scale images of black holes provide a potential probe of fundamental physics, including tests of gravity and black-hole hair. To assess the impact of scalar hair on accretion-flow imaging self-consistently, we construct an analytical model of a geometrically thick, magnetized disk around a rotating hairy black hole in Horndeski theory and analyze its 230 GHz image morphology. We find that scalar hair modestly alters the inflow and magnetic-field structure but strengthens gravitational redshift, markedly reducing the total flux and lensed ring brightness through relativistic transfer and spectral-shift effects. Moreover, we identify a novel, spatially resolved imaging feature: the maximum interferometric diameter of the first photon ring responds strongly to the hair parameter but shows little dependence on accretion-flow details, making it a promising observable for constraining black-hole hair with future space-based interferometry.