arXiv:2605.30770v2 Announce Type: replace Abstract: Advancing humanoid traversal toward the open world is key to practical deployment in human environments, but remains challenging. The robot must use vision to ensure safe and reliable foot placement on heterogeneous terrain during highly dynamic motion, while producing coordinated, natural whole-body behaviors. We propose SSR, an efficient end-to-end framework for egocentric vision-based humanoid traversal that jointly learns these capabilities. SSR introduces imagined foothold guidance, which learns to model forthcoming swing-foot contacts and evaluates their support to guide pre-touchdown swings toward stable regions, reducing edge slips. It further employs equivariant latent-space symmetry augmentation to efficiently induce bilateral coordination under high-dimensional visual observations, and uses terrain-specific multi-discriminator motion priors to encourage human-like behavior across scenes. Extensive experiments show that SSR achieves safe, stable, and high-quality locomotion on diverse real-world terrains, including stairs with varied structures and extreme challenges such as wide gaps and high platforms, while enabling reliable long-horizon traversal in diverse outdoor environments. Project website: https://ssr-humanoid.github.io/.