Coral skeletal growth is sensitive to environmental change and may be adversely impacted by an acidifying ocean. However, physiological processes can also buffer biomineralization from external conditions, providing apparent resilience to acidification in some species. These same physiological processes affect skeletal composition and can impact paleoenvironmental proxies. Understanding the mechanisms of coral calcification is thus crucial for predicting the vulnerability of different corals to ocean acidification and for accurately interpreting coral-based climate records. Here, using boron isotope (δ11B) measurements on cultured cold-water corals, we explain fundamental features of coral calcification and its sensitivity to environmental change. Boron isotopes are one of the most widely used proxies for past seawater pH, and we observe the expected sensitivity between δ11B and pH. Surprisingly, we also discover that coral δ11B is independently sensitive to seawater dissolved inorganic carbon (DIC). We can explain this new DIC effect if we introduce boric acid diffusion across cell membranes as a new flux within a geochemical model of biomineralization. This model independently predicts the sensitivity of the δ11B-pH proxy, without being trained to these data, even though calcifying fluid pH (pHCF) is constant. Boric acid diffusion resolves why δ11B is a useful proxy across a range of calcifiers, including foraminifera, even when calcifying fluid pH differs from seawater. Our modeling shows that δ11B cannot be interpreted unequivocally as a direct tracer of pHCF. Constant pHCF implies similar calcification rates as seawater pH decreases, which can explain the resilience of some corals to ocean acidification. However, we show that this resilience has a hidden energetic cost such that calcification becomes less efficient in an acidifying ocean