Scale2Real: Dimensionless Physical Representations for Cross-Scale Humanoid Locomotion

Full-scale humanoid learning is constrained by the cost, risk, and throughput of physical interaction, yet smaller embodiments are useful only if the experience they generate remains relevant to the deployment robot. We ask whether physical data-acquisition scale can be separated from deployment scale by preserving the dimensionless structure of locomotion dynamics. Using a controlled family of geometrically scaled H1 humanoids in MuJoCo/MJX, we compare raw dimensional observations with a physically scale-normalized representation across multiple training and evaluation scales. Raw policies lose full-scale transfer rapidly as the source embodiment is miniaturized: at $s=0.25$, normalized transfer to the full-scale target is $0.066\pm0.017$. With dimensionless physical normalization, the corresponding transfer is $1.034\pm0.196$, and the mean transfer gain is positive at every smaller confirmatory source scale. At a fixed budget of 6,553,600 full-scale target steps, selected normalized source policies yield higher endpoint return than full-scale scratch training in all three seeds. Together, these results show that representation choice materially contributes to the cross-scale degradation observed in this controlled scale family, rather than that degradation being an unavoidable consequence of geometric scale change. They motivate treating physical scale as a design variable for embodied data acquisition, with compute, sensing, latency, and hardware constraints determining how far miniaturization can be pushed.