Enzymes speed up biochemical reactions at the core of life by as much as 15 orders of magnitude. Yet, despite considerable advances, the fine dynamical determinants at the microscopic level of their catalytic proficiency are still elusive. In this work, we use a powerful mathematical approach to show that rate-promoting vibrations in the picosecond range, specifically encoded in the 3D protein structure, are localized vibrations optimally coupled to the chemical reaction coordinates at the active site. Remarkably, our theory also exposes an hithertho unknown deep connection between the unique localization fingerprint and a distinct partition of the 3D fold into independent, foldspanning subdomains that govern long-range communication. The universality of these features is demonstrated on a pool of more than 900 enzyme structures, comprising a total of more than 10,000 experimentally annotated catalytic sites. Our theory provides a unified microscopic rationale for the subtle structure-dynamics-function link in proteins.