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Pooled single-cell perturbation screens represent powerful experimental platforms for functional genomics, yet interpreting these rich datasets for meaningful biological conclusions remains challenging. Most current methods fall at one of two extremes: either opaque deep learning models that obscure biological meaning, or simplified frameworks that treat genes as isolated units. As such, these approaches overlook a crucial insight: gene co-fluctuations in unperturbed cellular states can be harnessed to model perturbation responses. Here we present CIPHER (Covariance Inference for Perturbation and High-dimensional Expression Response), a framework leveraging linear response theory from statistical physics to predict transcriptome-wide perturbation outcomes using gene co-fluctuations in unperturbed cells. We validated CIPHER on synthetic regulatory networks before applying it to 11 large-scale single-cell perturbation datasets covering 4,234 perturbations and over 1.36M cells. CIPHER robustly recapitulated genome-wide responses to single and double perturbations by exploiting baseline gene covariance structure. Importantly, eliminating gene-gene covariances, while retaining gene-intrinsic variances, reduced model performance by 11-fold, demonstrating the rich information stored within baseline fluctuation structures. Moreover, gene-gene correlations transferred successfully across independent experiments of the same cell type, revealing stereotypic fluctuation structures. Furthermore, CIPHER outperformed conventional differential expression metrics in identifying true perturbations while providing uncertainty-aware effect size estimates through Bayesian inference. Finally, most genome-wide responses propagated through the covariance matrix along approximately three independent and global gene modules. CIPHER underscores the importance of theoretically-grounded models in capturing complex biological responses, highlighting fundamental design principles encoded in cellular fluctuation patterns.