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Cellular diversity is a product of evolution acting to establish divergent gene regulatory programs from a common genome. Here, we use cross-cell-type epigenetic conservation to gain insight into the impact of selective constraints on genome function and phenotypic variation. By comparing chromatin accessibility across hundreds of diverse cell-types, we identify 1.4% of the human genome safeguarded by conserved domains of facultative heterochromatin, which we term regions under "cellular constraint". We calculate single-base resolution cellular constraint scores and demonstrate robust prediction of functionally important coding and noncoding loci in a cell-type-, trait-, and disease-agnostic manner. Cellular constraint annotation enhances causal variant identification, drug discovery, and clinical diagnostic predictions. Furthermore, cell-constrained sequences share signals of positive and negative selection, suggesting a dynamic role in influencing human traits and cellular phenotypes. Overall, this study demonstrates that evolutionary chromatin dynamics can be leveraged to inform the translation of genetic discoveries into effective biological, therapeutic, and clinical outcomes.