Lossy compression is essential for managing massive scientific data, but per-element error bounds do not translate into bounds on downstream quantities of interest (QoIs) such as regional averages, neural network predictions, or multi-field derived quantities. We present TOPIQ, a statistical error-propagation framework that predicts QoI-level bias and uncertainty from compact compression metadata (less than 0.1% of original data). TOPIQ decomposes QoIs into primitive operators with closed-form propagation rules accounting for spatial error correlation and data-error coupling; new QoIs are supported by composition at runtime with no per-QoI derivation or retraining. Across 552 evaluations spanning 4 datasets, 3 compressors, 4 QoI families, and 8 error bounds, 93.1% of configurations achieve well-calibrated predictions. Pre-computed metadata enables post-hoc uncertainty quantification for arbitrary query regions at 56x-402x speedup over direct computation. A case study demonstrates integration into an AI-driven analysis pipeline with end-to-end confidence intervals for dynamically composed queries.

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