Data processing systems are a fundamental component of the modern computing stack. These systems are routinely deployed online: they continuously receive the requests of data processing operations, and continuously return the results to end users or client applications. Online data processing systems have unique features beyond conventional data processing, and the optimizations designed for them are complex, especially when data themselves are structured and dynamic. This paper describes DON Calculus, the first rigorous foundation for online data processing. It captures the essential behavior of both the backend data processing engine and the frontend application, with the focus on two design dimensions essential yet unique to online data processing systems: incremental operation processing (IOP) and temporal locality optimization (TLO). A novel design insight is that the operations continuously applied to the data can be defined as an operation stream flowing through the data structure, and this abstraction unifies diverse designs of IOP and TLO in one calculus. DON Calculus is endowed with a mechanized metatheory centering around a key observable equivalence property: despite the significant non-deterministic executions introduced by IOP and TLO, the observable result of DON Calculus data processing is identical to that of conventional data processing without IOP and TLO. Broadly, DON Calculus is a novel instance in the active pursuit of providing rigorous guarantees to the software system stack. The specification and mechanization of DON Calculus provide a sound base for the designers of future data processing systems to build upon, helping them embrace rigorous semantic engineering without the need of developing from scratch.