Discontinuous Prior-Mode Sections and the Geometry of Ambiguity in Intrinsic Image Decomposition

The viral 2015 photograph known as "The Dress" divides observers into two camps because it is ambiguous: the same image colors can be explained either as a blue-black surface under one illuminant or as a white-gold surface under another. We propose a geometric account in which the ambiguity arises from a singularity in intrinsic image decomposition, the inverse problem of separating an observed image into reflectance and illumination. Our central claim is that the prior-mode section, i.e. the prior-preferred decomposition, switches across an ambiguity boundary in image space, and that any smooth learned model can only approximate this discontinuous switch by forming a thin transition layer. This predicts two observable signatures, where $\Delta$ is the jump between branches of the prior-mode section and $\lambda$ is the regularization strength: for inverse decomposers, an albedo Jacobian scaling as $|\Delta|/\sqrt{\lambda}$; and for forward encoders, the Fernet curvature that blows up on a scale of $1/\sqrt{\lambda}$. On CGIntrinsics ($N=1998$ images, $n=2\times 10^7$ pixels), the color-temperature albedo Jacobian of Careaga DPT has partial Spearman correlation $r=0.41$ with dense ground-truth albedo error, compared with $r=0.087$ and $r=0.021$ for brightness and saturation controls. On "The Dress", CLIP ViT-L/14 exhibits a latent curvature peak of $\kappa=73.03$ at $6473\,\mathrm{K}$, one sampled step from D65 daylight, while a control dress image peaks at $\kappa=34.75$ with no comparable feature near D65. The same characteristic appears across architectures (U-Net inverter, diffusion inverter, ViT encoder) and datasets (rendered indoor scenes, web photograph), each measured with the observable appropriate to its model class.