A two-point curve neighborhood is the union of degree d rational curves that pass through two given Schubert varieties in the flag variety. These varieties were introduced by Buch-Chaput-Mihalcea-Perrin in 2013. They encode information about the 3-point Gromov-Witten invariants of the flag variety. Postnikov proved there is a unique minimal degree where the curve neighborhood is non-empty and described it using the quantum Bruhat graph. In this case, we provide a concrete description of the curve neighborhood in terms of the combinatorics of tilted Bruhat order and a stratification with open tilted Richardson varieties. This is joint work with Jiyang Gao and Yibo Gao.