Everything at Every Scale: Scale-Invariant Diffusion with Continuous Super-Resolution

Creating images from noise is image generation; reconstructing fine details from coarse inputs is super-resolution. Despite their practical differences, both can be understood as reversing information loss across scales. We introduce **SKILD**, a **S**cale-invariant **K**-Space **I**mage **L**earning **D**iffusion model that unifies generation and continuous super-resolution within a single unconditional framework. Both natural images and critical physical systems exhibit scale invariance, and we leverage it to design a forward process that attenuates image content from fine to coarse scales while injecting spectrum-matched Gaussian noise, making scale an explicit coordinate of the diffusion dynamics. The same trained reverse process performs generation and continuous super-resolution by varying only the starting timestep: *no task-specific architecture, no conditioning branch, no classifier-free guidance, no retraining per scale factor*. Empirically, SKILD reaches FID $2.65$ and Inception Score $9.63$ on unconditional CIFAR-10, performs $2\times$–$8\times$ super-resolution on ImageNet from a single unconditional checkpoint while outperforming conditional models across perceptual metrics, and reconstructs critical Ising models whose connected four-point correlations closely track the ground truth.