Electron pairing at low temperatures leads to superconductivity. A fundamental question is whether more complex states—characterized by order in four-electron composite objects, termed electron quadrupling or composite order—can exist in materials, and if so, under what conditions they emerge and what properties they exhibit. These states lie beyond the scope of Bardeen-Cooper-Schrieffer theory, and a microscopic description of them remained elusive. In the first part of the paper, we provide a general microscopic framework to describe these and the other four-fermion composite states. In the second part of the paper, we derive and solve a specific fermionic model in two and three dimensions that hosts time-reversal symmetry-breaking electron quadrupling order. The fermionic microscopic theory is used to estimate the specific heat and electron density of states.