The Determinism of Randomness: Latent Space Degeneracy in Diffusion Model

arXiv:2511.07756v4 Announce Type: replace Abstract: Diffusion models initialize generation from an isotropic Gaussian latent, yet changing only the random seed can substantially alter prompt faithfulness, composition, and visual quality. We explain this gap by distinguishing the Euclidean geometry of the prior from the semantic geometry induced by the sampler: the effective map from initial noise to semantic outcome has many semantic-invariant directions and a much smaller set of semantic-sensitive directions. This induces a degenerate pullback semi-metric on the latent space and provides a geometric view of the seed lottery. Guided by this view, we propose a training-free inference procedure that estimates a prompt-aligned horizontal proxy from a single high-noise cold-start probe and applies tangential seed injection followed by spherical retraction to remain on the prior's typical shell. Across image, video, and 3D generation benchmarks, the method improves alignment and quality metrics over standard sampling.