Phase-space entropy at acquisition reflects downstream learnability

Abstract: Modern learning systems work with data that vary widely across domains, but they all ultimately depend on how much structure is already present in the measurements before any model is trained. This raises a basic question: is there a general, modality-agnostic way to quantify how acquisition itself preserves or destroys the information that downstream learners could use? Here we propose an acquisition-level scalar $ΔS_{\mathcal B}$ based on instrument-resolved phase space. Unlike pixelwise distortion or purely spectral errors that often saturate under aggressive undersampling, $ΔS_{\mathcal B}$ directly quantifies how acquisition mixes or removes joint space--frequency structure at the instrument scale. We show theoretically that (ΔS_{\mathcal B}) correctly identifies the phase-space coherence of periodic sampling as the physical source of aliasing, recovering classical sampling-theorem consequences. Empirically, across masked image classification, accelerated MRI, and massive MIMO (including over-the-air measurements), $|ΔS_{\mathcal B}|$ consistently ranks sampling geometries and predicts downstream reconstruction/recognition difficulty \emph{without training}. In particular, minimizing $|ΔS_{\mathcal B}|$ enables zero-training selection of variable-density MRI mask parameters that matches designs tuned by conventional pre-reconstruction criteria. These results suggest that phase-space entropy at acquisition reflects downstream learnability, enabling pre-training selection of candidate sampling policies and as a shared notion of information preservation across modalities.