Uncertainty as an Underconstrained Axis in Lossy Compression

Classical rate-distortion theory minimises expected distortion at a given coding rate but does not constrain the uncertainty of the reconstructed signal at the decoder. We formalise predictive uncertainty $U$ as the conditional variance of the reconstruction given the source, and show that it is an underconstrained degree of freedom of RD-optimal compression with a sharp empirical signature: at matched rate and matched scalar $U$, two reconstructions of the same image can differ by $40$+ mIoU points in downstream segmentation depending on where the variance is spatially allocated, with $88\%$ of (image, quality) pairs showing a significantly negative within-image slope of mIoU on $U$. We derive a closed-form Uncertainty-Rate-Distortion surface and prove a coding-theoretic converse for additive-noise channels with linear regression under Gaussian assumption. A region-importance spatial-URD functional also converts the spatial-allocation finding into a within-image predictor that generalises across two segmentation architectures, three rates, two codecs, and a different segmentation foundation model on a different dataset. Cross-task and cross-modality probes (BLIP captioning, CLIP zero-shot, NYU depth, LibriSpeech speech recognition) characterise the framework's reach: the spatial-URD prediction transfers cleanly to locally-aggregating downstream models on both image and audio modalities, and degenerates predictably for globally-aggregating ones. The validation code and a demo for a diffusion task is provided as Supp. Mat., and will be released alongside with all the code to reproduce the experiments upon acceptance of the work.