Noising Creates Nothing, Destroys Nothing: On the Identifiability of Conditional Diffusion Models

Diffusion models have a built-in time variable $t$ that indexes a systematically varying, known noise level. This invites the natural hypothesis that $t$ can serve as a segment index or auxiliary variable for identifiability, as in the time-contrastive learning of Hyvärinen & Morioka 2016. We study this conjecture for observation-space conditional diffusion models whose conditioning acts through the latent prior, and show why it fails. Diffusion time changes the observation channel but leaves the underlying ICA source distribution unchanged, so the latent-side nonstationarity exploited by time-contrastive learning is absent. Folding $t$ into a genuine auxiliary variable $u$ does not enlarge the diversity supplied by $u$. Moreover, the forward parameterization indexed by $(t,u)$ falls outside the model class of Khemakhem et al. 2020, and no reparameterization repairs this. At any fixed noise level, if $u$ is present, the theorem of Khemakhem et al. 2020 applies under the standard regularity and diversity assumptions of auxiliary-variable nonlinear ICA and identifies the model up to its linear equivalence relation. Thus noising creates no identifiability on its own, while the identifiability already supplied by $u$ remains available at fixed $t$, for both the variance-preserving (VP) and variance-exploding (VE) forward processes of Song et al. 2021.