We show that the supremum of a canonical process under a given marginal index distribution can be characterized by an integral of the rate-distortion function of the index distribution across various scales. This gives a sharp two-sided version of Dudley's integral by replacing the log covering number by the rate-distortion function from information theory. As applications, we provide new streamlined proofs of the majorizing measure theorem and the Bednorz-Latala theorem (formerly the Bernoulli conjecture). Furthermore, the value of entropic optimal transport can be sharply characterized by a truncated rate-distortion integral. (Based on arXiv:2508.18682, 2604.14061, and 2608.11031).