A Simplex Witness Certificate for Constant Collapse in Variational Autoencoders

This note studies exact constant collapse in variational autoencoders, where the encoder mean becomes independent of the input. The goal is to make this specific failure mode pre-designable, monitorable during training, and certifiable after training. The prior is kept as the standard Gaussian. Given a fixed teacher posterior, we attach to the latent mean a fixed simplex witness head. The resulting teacher-student alignment loss has an exact constant-predictor baseline equal to the teacher information. If the alignment loss is below this baseline, the latent mean cannot be input-independent constant collapsed. The simplex witness also has a closed-form inverse. Any full-support teacher posterior can be represented by embedding its centered log-odds into the latent space. This gives an explicit latent energy cost and explains when the alignment loss can be made small. A computable view gap handles the case where teacher targets are computed from a different view. Thus exact constant collapse is converted from an after-the-fact training pathology into a design-and-certificate problem.