Diffusion Likelihoods Need Certificates: Two Reported Values Not Established as Discrete Upper Bounds

Diffusion, flow and discrete-diffusion models are ranked by negative log-likelihood in bits per dimension, and the reported numbers are read as upper bounds on the negative log-probability of discrete data. A reported number is, however, a finite evaluation: a Monte Carlo estimate over noise levels, at a finite endpoint, sometimes of a continuous density. We show that this gap bears on two recent headline results. For QDPM (0.27 bits/dim on CIFAR-10), under unit-scale data encoding and a gate applied to a finite float32 network output, the paper's own bound (its Eq. 19) is at least 3.80 bits/dim for every network and every 8-bit image, and 5.70 bits/dim on the CIFAR-10 test set; 0.27 is not its value. For Warm Start (2.50 bits/dim), the reported total at its endpoint is a continuous Gaussian cross-entropy plus a score-matching integral, with no dequantization term; on the uniform law over the 256 data levels (entropy 8 bits) the same two terms with the exact score total 6.90 bits, so at this endpoint the total is not a discrete upper bound in general. Each mechanism follows in a page from known tools: the I-MMSE relation, the order statistics of noise-level sampling, and the dequantization inequality. We propose a certificate protocol assembled from these tools, so that a reported likelihood states which quantity it bounds and with what confidence.